Fractional Hermite–Hadamard, Newton–Milne, and Convexity Involving Arithmetic–Geometric Mean-Type Inequalities in Hilbert and Mixed-Norm Morrey Spaces ℓ_q(·)(M_p(·),v(·)) with Variable Exponents

Abstract

Function spaces play a crucial role in the study and application of mathematical inequalities. They provide a structured framework within which inequalities can be formulated, analyzed, and applied. They allow for the extension of inequalities from finite-dimensional spaces to infinite-dimensional contexts, which is crucial in mathematical analysis. In this note, we develop various new bounds and refinements of different well-known inequalities involving Hilbert spaces in a tensor framework as well as mixed Moore norm spaces with variable exponents. The article begins with Newton–Milne-type inequalities for differentiable convex mappings. Our next step is to take advantage of convexity involving arithmetic–geometric means and build various new bounds by utilizing self-adjoint operators of Hilbert spaces in tensorial frameworks for different types of generalized convex mappings. To obtain all these results, we use Riemann–Liouville fractional integrals and develop several new identities for these operator inequalities. Furthermore, we present some examples and consequences for transcendental functions. Moreover, we developed the Hermite–Hadamard inequality in a new and significant way by using mixed-norm Moore spaces with variable exponent functions that have not been developed previously with any other type of function space apart from classical Lebesgue space. Mathematical inequalities supporting tensor Hilbert spaces are rarely examined in the literature, so we believe that this work opens up a whole new avenue in mathematical inequality theory.

1. Introduction

Fractional calculus is a fascinating branch of mathematics that extends the familiar concepts of differentiation and integration to noninteger orders. Fractional derivatives and integrals possess a number of key properties, including linearity, the Leibniz rule, and composition rules, which make them useful for analyzing and manipulating fractional differential and integral equations. Fractional calculus has numerous intriguing aspects, such as nonlocality, where the fractional derivative is produced by integrating over a whole range of values, and there is a nontrivial dependence on the integration’s lower bound. Some applications of fractional calculus include the following: Fractional derivatives incorporate past information, making them suitable for modeling systems with memory effects. Fractional calculus is used in electrochemical kinetics to relate the substrate concentration to the redox behavior of substrates in a solution. In 2013–2014, Atangana et al. [1] used fractional derivatives to express groundwater flow problems; to address definitional inconsistencies, some problems are specified using fractional integrals. Regarding some other recent uses in several applied science domains, see refs. [2,3,4,5,6,7,8,9,10,11,12,13,14,15].

Operator fractional integral inequalities are essential to many areas of mathematics and their associated applications. They include the comparison of operators or matrices using various norms, traces, and determinants. The analysis and solution of fractional integral and differential equations require the use of fractional integral inequalities. In probability theory, fractional integral inequalities are used to investigate stochastic processes and their properties. Fractional integral inequalities are also essential in functional analysis, especially when studying function spaces with fractional order norms and operators. A deeper understanding of fractional differential equations can be gained through the analysis of fractional integral inequalities in convex analysis using techniques from fractional calculus, generalized convexity, and by using various integral identities. In many mathematical applications, it is helpful to be able to represent the integral of a convex function in terms of bounds or estimations.

A number of significant integral inequalities have been reported in the literature, including Simpson, Bullen, Ostrowski, Trapezoid, Jensen, midpoint, and many others. Several relevant inequalities have been derived for differentiable convex functions [16], Lipschitzian functions [17], bounded functions [18], functions of bounded variation [19], and twice differentiable convex functions [20]. The British mathematician William John Milne (1843–1914) proposed Milne-type inequalities. It has been found that these inequalities have applications in a wide variety of mathematics fields, including special function analysis [21], approximation theory [22], and numerical analysis [23]. They are useful for bounding mistakes and analyzing the correctness of numerical and analytical methods because they use integrals to quantify the difference between a function and its approximation. Different authors investigate Milne-type inequalities in different settings. For example, in [24], the authors used interval-valued mapping and inclusion relations; in [25], quantum integrals were used to find various bounds; in [26], the authors used differentiable convex functions by using fractional operators; and in [27], the authors introduced some new identities and illustrated some applications to means. Suppose that $\mathsf{\vartheta }:[{\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2]\to \mathsf{R}$ is a four-times continuously differentiable mapping on $({\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2)$, and let ${∥{\mathsf{\vartheta }}^{\left(4\right)}∥}_{\infty }={sup}_{\mathsf{\kappa }\in \left({\mathsf{\varpi }}_1{\mathsf{\varpi }}_2\right)}\left|{\mathsf{\vartheta }}^{\left(4\right)}\left(\mathsf{\kappa }\right)\right|<\infty$. Then, one has the following double inequality [28]:

$$\left|{\displaystyle \frac{1}{3}}\left[2\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)-\mathsf{\vartheta }\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right)+2\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)\right]-{\displaystyle \frac{1}{{\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1}}{\int }_{{\mathsf{\varpi }}_1}^{{\mathsf{\varpi }}_2}\mathsf{\vartheta }\left(\mathsf{\kappa }\right)\mathtt{d}\mathsf{\kappa }\right|\le {\displaystyle \frac{7{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^4}{23040}}{∥{\mathsf{\vartheta }}^{\left(4\right)}∥}_{\infty }.$$

Recently, operators with convexity and different types of norm structures have been used to develop these types of inequalities. The notion of tensor Hilbert spaces refers to a generalization of Hilbert spaces that include tensors and are used to represent multilinear mappings between Hilbert spaces. Functional analysis relies heavily on tensor inequalities and multilinear maps between Hilbert spaces. Inequalities of the tensor Hilbert space are fundamental to functional analysis and quantum mechanics, but they are seldom discussed and generalized in different contexts. However, some recent advances have led to the following results. Markus Weimar [29] conceptualized linear problems on Hilbert spaces in terms of tensor products and devised a linear algorithm based on eigenvalues of a continuous linear function. Lewintan [30] uses conformally invariant dislocation energy to show some refinements and reverses for Korn inequalities in three-dimensional tensorial space. J. C Dunn [31] discusses a variety of new bounds and various properties of different types of generalized convex mappings, monotonicity, and gradient processes in Hilbert space. Mohammad and Bakherad [32] establish many operator extensions of the Chebyshev inequality, the primary one involving the Hadamard product of Hilbert space operators. Krnić et al. [33] used joint concavity and multidimensional weighted geometric means to construct Jensen’s inequality on Hilbert space. In [34], the authors examine the ill-posedness and stability of the economic equilibrium model under tensor variational inequalities. S. S. Dragomir [35] used the tensorial and Hadamard product to develop several new inequalities for synchronous functions. Prasenjit and Tapas [36] examine frames in the tensor product of n-Hilbert spaces using several forms of sequences. For further detail on these types of inequalities, see [37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55] and the references therein.

Theorem 1 

(see [36]). Let ${\left\{{\mathtt{r}}_{\mathtt{i}}\right\}}_{\mathtt{i}=1}^{\infty }$ and ${\left\{{\mathtt{s}}_{\mathtt{j}}\right\}}_{\mathtt{j}=1}^{\infty }$ be the two sequences in Hilbert spaces $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$. Then, for each $\mathtt{r}\otimes \mathtt{s}\in$ $\mathsf{{\rm Y}}\otimes \mathsf{\zeta }-\{\mathsf{\vartheta }\otimes \mathsf{\vartheta }\}$, there exist constants ${\mathsf{\kappa }}_1,{\mathsf{\kappa }}_2>0$ such that

$$\begin{array}{cc}\hfill {\mathsf{\kappa }}_1{∥\mathtt{r}\otimes \mathtt{s},{\mathtt{u}}_{\mathtt{2}}\otimes {\mathtt{v}}_{\mathtt{2}},\cdots ,{\mathtt{u}}_{\mathtt{n}}\otimes {\mathtt{v}}_{\mathtt{n}}∥}^2& \le \sum _{\mathtt{i},\mathtt{j}=1}^{\infty }{\left|〈\mathtt{r}\otimes \mathtt{s},{\mathtt{r}}_{\mathtt{i}}\otimes {\mathtt{s}}_{\mathtt{j}}\mid {\mathtt{u}}_{\mathtt{2}}\otimes {\mathtt{v}}_{\mathtt{2}},\cdots ,{\mathtt{u}}_{\mathtt{n}}\otimes {\mathtt{v}}_{\mathtt{n}}〉\right|}^2\hfill \\ \hfill & \le {\mathsf{\kappa }}_2{∥\mathtt{r}\otimes \mathtt{s},{\mathtt{u}}_{\mathtt{2}}\otimes {\mathtt{v}}_{\mathtt{2}},\cdots ,{\mathtt{u}}_{\mathtt{n}}\otimes {\mathtt{v}}_{\mathtt{n}}∥}^2.\hfill \end{array}$$

Stojiljković et al. [56] developed novel Trapezoid-type inequalities utilizing differentiable convex mappings on Hilbert spaces.

Theorem 2 

(see [56]). Let the self-adjoint operators $\mathsf{\sigma }$ and $\mathsf{\rho }$ have $\mathtt{Sp}\left(\mathsf{\sigma }\right)\subset \mathsf{\Omega }$ and $\mathtt{Sp}\left(\mathsf{\rho }\right)\subset$ Ω. Assume that $\mathsf{\vartheta }$ is a continuous and differentiable convex mapping on Ω; then, one has

$$\begin{array}{cc}& 0⩽(1-\mathsf{\delta })\mathsf{\vartheta }\left({\displaystyle \frac{\mathsf{\rho }}{\mathsf{\sigma }}}\right)\otimes 1+\mathsf{\delta }1\otimes \mathsf{\vartheta }\left({\displaystyle \frac{\mathsf{\delta }}{\mathsf{\sigma }}}\right)-\mathsf{\vartheta }\left({\displaystyle \frac{1-\mathsf{\delta }}{\mathsf{\sigma }}}\mathsf{\rho }\otimes 1+\mathsf{\delta }1\otimes \mathsf{\delta }\right)\hfill \\ & +\mathsf{\delta }\left(1-{\displaystyle \frac{1}{\mathsf{\sigma }}}\right){\int }_0^1(1\otimes \mathsf{\delta }){\mathsf{\vartheta }}^{\prime }\left({\displaystyle \frac{1-\mathsf{\delta }\mathtt{u}}{\mathsf{\sigma }}}\mathsf{\rho }\otimes 1+\mathtt{u}\mathsf{\delta }1\otimes \mathsf{\delta }\right)\mathtt{d}\mathtt{u}\hfill \\ & ⩽\left({\displaystyle \frac{\mathsf{\delta }(1-\mathsf{\delta })}{\mathsf{\sigma }}}\mathsf{\rho }\otimes 1-\mathsf{\delta }·\left({\displaystyle \frac{1}{\mathsf{\sigma }}}-\mathsf{\delta }\right)1\otimes \mathsf{\delta }\right)\left({\mathsf{\vartheta }}^{\prime }\left(\mathsf{\rho }\right)\otimes 1-1\otimes {\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }\right)\right).\hfill \end{array}$$

Shuhei Wada generalized the Cauchy–Schwarz operator matrix inequality on Hilbert spaces as follows:

Theorem 3 

(see [57]). Assume that on a Hilbert space, $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ are positive semidefinite operators. Then,

$$\begin{array}{cc}\hfill (\mathsf{{\rm Y}}\#\mathsf{\zeta })\otimes (\mathsf{{\rm Y}}\#\mathsf{\zeta })& ⩽{\displaystyle \frac{1}{2}}\left\{\left(\mathsf{{\rm Y}}\mathsf{\sigma }\mathsf{\zeta }\right)\otimes \left(\mathsf{{\rm Y}}{\mathsf{\sigma }}^{\perp }\mathsf{\zeta }\right)+\left(\mathsf{{\rm Y}}{\mathsf{\sigma }}^{\perp }\mathsf{\zeta }\right)\otimes \left(\mathsf{{\rm Y}}\mathsf{\sigma }\mathsf{\zeta }\right)\right\}\hfill \\ & ⩽{\displaystyle \frac{1}{2}}\{(\mathsf{{\rm Y}}\otimes \mathsf{\zeta })+(\mathsf{\zeta }\otimes \mathsf{{\rm Y}})\}.\hfill \end{array}$$

Dragomir [58] studied a number of novel and significant tensorial perspective properties for convex functions on Hilbert spaces using various kinds of differentiable convex mappings.

Theorem 4 

(see [58]). Assume that $\mathsf{\vartheta }$ is convex on $(0,\infty ),0<{\mathsf{\kappa }}_1\le \mathtt{G},\mathtt{H}\le {\mathsf{\kappa }}_2$ for some constants ${\mathsf{\kappa }}_1,{\mathsf{\kappa }}_2$, and $\mathsf{\phi }\in \partial \mathsf{\vartheta }$; then,

$$\begin{array}{cc}\hfill {\mathtt{P}}_{\mathsf{\vartheta },\otimes }(\mathtt{G},\mathtt{H})& \ge \left[\mathsf{\vartheta }\left({\displaystyle \frac{{\mathsf{\kappa }}_{\mathtt{1}}+{\mathsf{\kappa }}_{\mathtt{2}}}{2}}\right)-\mathsf{\phi }\left({\displaystyle \frac{{\mathsf{\kappa }}_{\mathtt{1}}+{\mathsf{\kappa }}_{\mathtt{2}}}{2}}\right){\displaystyle \frac{{\mathsf{\kappa }}_{\mathtt{1}}+{\mathsf{\kappa }}_{\mathtt{2}}}{2}}\right](1\otimes \mathsf{\zeta })\hfill \\ & +\mathsf{\phi }\left({\displaystyle \frac{{\mathsf{\kappa }}_{\mathtt{1}}+{\mathsf{\kappa }}_{\mathtt{2}}}{2}}\right)(\mathsf{{\rm Y}}\otimes 1).\hfill \end{array}$$

Lebesgue spaces with variable exponents, represented as ${\mathtt{L}}_{\mathtt{p}}(·)$, are an extension of the traditional Lebesgue spaces ${\mathtt{L}}_{\mathtt{p}}$, in which the exponent $\mathtt{p}$ is subject to variation based on the point $\mathtt{x}$ inside the domain. The analysis of hydrodynamic equations that describes the behavior of non-Newtonian fluids naturally gives birth to Lebesgue spaces with a variable exponent. Analyzing solutions to elliptic equations with variable coefficients of partial differential equations as well as investigating features of analytic functions with varying growth behavior are some common examples of these analyses; see [59,60] and the references therein.

Variable exponent Lebesgue spaces were first investigated as Banach spaces in [61], although they initially appeared in [62]. Furthermore, modular function spaces are deeply linked with generalized Lebesgue spaces, which are two fascinating and complementing aspects of modern functional analysis. It provides a versatile toolkit for analyzing functions and their interactions across various domains when studying modular function spaces in combination with other different spaces. A wide variety of inequalities have been developed through the investigation of modular function spaces coupled with the following spaces with varying exponents, including Sobolev spaces [63], Morrey spaces [64], Herz spaces [65], Campanato spaces [66], Hardy spaces [67], Grand Lebesgue spaces [68], sequence Lebesgue spaces [69], and various others.

Somayeh [70] studied several functional inequalities in the context of variable exponent functions. The following is a weighted Hardy-type inequality and is defined as follows:

Theorem 5 

(see [70]). Consider three positive weight functions, $\mathtt{w},\mathtt{v}$, and $\mathtt{p}:(0,\infty )⟶(0,\infty )$ such that $0<$ ${\mathtt{p}}^{-}\le {\mathtt{p}}^{+}<\infty ,where{\mathtt{p}}^{+}=\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}}}{ess}sup\mathtt{p}\left(\sigma \right),{\mathtt{p}}^{-}=\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}}}{ess\; inf}\mathtt{p}\left(\sigma \right)$. If some positive constant $\mathtt{M}$ exists,

$${\int }_0^{\infty }{\left({\displaystyle \frac{1}{\mathtt{s}}}{\int }_0^{\mathtt{s}}\mathsf{\vartheta }\left(\mathtt{t}\right)\mathtt{d}\mathtt{t}\right)}^{\mathtt{p}\left(\mathtt{s}\right)}\mathtt{v}\left(\mathtt{s}\right)\mathtt{ds}\le \mathtt{M}{\int }_0^{\infty }\mathsf{\vartheta }{\left(\mathtt{s}\right)}^{\mathtt{p}\left(\mathtt{s}\right)}\mathtt{w}\left(\mathtt{s}\right)\mathtt{ds}.$$

Sultan et al. [71] demonstrate the boundedness of the Riesz potential operator on Herz–Morrey spaces with varying exponents.

Theorem 6 

(see [71]). If $0<{\mathtt{s}}_{\mathtt{i}}<\infty ,1\le {\mathtt{q}}_{-}\le {\mathtt{q}}_{+}<\infty ,where{\mathtt{q}}_{+}=\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}}}{ess}sup\mathtt{q}\left(\sigma \right),{\mathtt{q}}_{-}=\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}}}{ess\; inf}\mathtt{q}\left(\sigma \right),\mathsf{\gamma }(·)\in {\mathtt{L}}^{\infty }\left({\mathsf{R}}^{\mathtt{n}}\right),{\displaystyle \frac{1}{\mathtt{s}}}={\displaystyle \frac{1}{{\mathtt{s}}_{\mathtt{1}}}}+{\displaystyle \frac{1}{{\mathtt{s}}_{\mathtt{2}}}}$, $1={\displaystyle \frac{1}{\mathtt{q}(·)}}+{\displaystyle \frac{1}{{\mathtt{q}}^{\prime }(·)}},\mathsf{\beta }={\mathsf{\beta }}_1+{\mathsf{\beta }}_2$, and $\mathsf{\gamma }(·)=\mathsf{\gamma }{(·)}_1(·)+{\mathsf{\gamma }}_2(·)$. Then,

$$\begin{array}{cc}\hfill & \underset{\Delta >0}{sup}\underset{\mathtt{k}\in \mathsf{Z}}{sup}{2}^{-\mathtt{k}\mathsf{\beta }}{\left({\Delta }^{\mathsf{\vartheta }}\sum _{\mathtt{k}\in \mathsf{Z}}{2}^{\mathtt{k}\mathsf{\beta }(·)\mathtt{s}(1+\mathsf{\beta })}{∥\mathsf{\varphi }{\mathsf{\chi }}_{\mathtt{k}}∥}_{{\mathtt{L}}^1}^{\mathtt{s}(1+\Delta )}\right)}^{{\displaystyle \frac{1}{\mathtt{s}(1+\Delta )}}}\hfill \\ & \le \mathtt{M}\underset{\Delta >0}{sup}\underset{\mathtt{k}\in \mathsf{Z}}{sup}{2}^{-\mathtt{k}\mathsf{\beta }}{\left({\Delta }^{\mathsf{\vartheta }}\sum _{\mathtt{k}\in \mathsf{Z}}{\left(2^{\mathtt{k}{\mathsf{\beta }}_1(·)}{∥\mathsf{\varphi }{\mathsf{\chi }}_{\mathtt{k}}∥}_{{\mathtt{L}}^{\mathtt{q}(·)}}\right)}^{{\mathtt{s}}_1(1+\Delta )}\right)}^{{\displaystyle \frac{1}{{\mathtt{s}}_1(1+\Delta )}}}\hfill \\ \hfill & \times \underset{\Delta >0}{sup}\underset{\mathtt{k}\in \mathsf{Z}}{sup}{2}^{-\mathtt{k}\mathsf{\beta }}{\left({\Delta }^{\mathsf{\vartheta }}\sum _{\mathtt{k}\in \mathsf{Z}}{\left(2^{\mathtt{k}{\mathsf{\beta }}_1(·)}{∥\mathsf{\varphi }{\mathsf{\chi }}_{\mathtt{k}}∥}_{{\mathtt{L}}^{{\mathtt{q}}^{\prime }(·)}}\right)}^{{\mathtt{s}}_2(1+\Delta )}\right)}^{{\displaystyle \frac{1}{{\mathtt{s}}_2(1+\Delta )}}}.\hfill \end{array}$$

We refer to [72,73] and the references therein for a few other relevant results and inequalities related to the problem examined in this note.

As a result of this study, we developed the Hermite–Hadamard inequality in a novel and significant manner by using mixed-norm Moore spaces with variable exponents, a concept that has not been developed recently with any type of function space. In addition, with remarks, we show that under the setting of certain types of exponent functions, we can recover some existing results. As a result of imposing some new specific conditions on exponent functions, this inequality refines Jensen and triangular inequalities. Moreover, we developed Newton–Milne inequalities and convexity involving arithmetic–geometric mean-type operator inequalities in Hilbert spaces by using some relational properties and different types of arithmetic operations from tensor analysis. Furthermore, we developed our main results using some interesting fractional identities as well as some examples that involved exponential- and logarithmic-type functions.

The works of these authors [56,58,70,71] inspire us in particular to develop a new and improved form of various inequalities in two different types of function spaces that is subject to several new refinements and reverses under different types of settings. Inequality theory is fundamentally enriched and expanded by the use of new methodologies and perspectives that have very rarely been discussed in a few articles. This opens up a new avenue in the area of inequality theory. This work is organized into eight sections, beginning with an introduction and basic discussion of the subject. In Section 3, we develop many significant identities and lemmas that are employed in the main discoveries. In Section 4, we use numerous significant fractional identities to build a Newton–Milne-type inequality for differentiable convex mappings. In Section 5, we use convexity with an arithmetic–geometric mean to establish double inequality. In Section 6, we present some examples and consequences for transcendental functions. In Section 7, we derive the Hermite–Hadamard inequality in mixed-norm Moore space with variable exponents. Finally, in Section 8, we present a precise conclusion and some future possibilities.

2. Preliminaries

In this note, we will discuss some fundamental concepts related to different types of function spaces, fractional calculus and their associated identities, and some relational features of Hilbert spaces. Some key concepts are not thoroughly discussed here, and thus we refer to [35].

Definition 1 

(see [74]). An inner product space is a normed vector space $\mathtt{X}$ on which an inner product is defined; that is, a mapping from $\mathtt{X}\times \mathtt{X}$ into its associated scalar field $\mathtt{K}$ that satisfies certain properties. An inner product space that is also complete is called a Hilbert space and is often denoted by $\mathtt{H}$. The inner product of two elements ${\mathsf{\eta }}_1,{\mathsf{\eta }}_2$ in $\mathtt{X}$ is denoted $\langle {\mathsf{\eta }}_1,{\mathsf{\eta }}_2\rangle .$ For all vectors ${\mathsf{\eta }}_1,{\mathsf{\eta }}_2,{\mathsf{\eta }}_3\in \mathtt{X}$ and scalars $\mathsf{\lambda }\in \mathtt{K}$, we have

$$\begin{array}{cc}\hfill \langle {\mathsf{\eta }}_1+{\mathsf{\eta }}_2,{\mathsf{\eta }}_3\rangle & =\langle {\mathsf{\eta }}_1,{\mathsf{\eta }}_3\rangle +\langle {\mathsf{\eta }}_2+{\mathsf{\eta }}_3\rangle \hfill \\ \hfill \langle \mathsf{\lambda }{\mathsf{\eta }}_1,{\mathsf{\eta }}_2\rangle & =\mathsf{\lambda }\langle {\mathsf{\eta }}_1,{\mathsf{\eta }}_2\rangle \hfill \\ \hfill \langle {\mathsf{\eta }}_1,{\mathsf{\eta }}_2\rangle & =\overline{\langle {\mathsf{\eta }}_2,{\mathsf{\eta }}_1\rangle }\hfill \\ \hfill \langle {\mathsf{\eta }}_1,{\mathsf{\eta }}_1\rangle & \ge 0,\langle {\mathsf{\eta }}_1,{\mathsf{\eta }}_1\rangle =0⟺{\mathsf{\eta }}_1=0.\hfill \end{array}$$

Definition 2 

(see [58]). A bilinear mapping $\mathsf{\vartheta }:\mathsf{{\rm Y}}\times \mathsf{\zeta }\to \mathtt{P}$ and a tensor product of $\mathsf{{\rm Y}}$ with $\mathsf{\zeta }$ provide a Hilbert space $\mathtt{P}$ such that

• The collection of all vectors $\mathsf{\vartheta }({\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2)({\mathsf{\varpi }}_1\in \mathsf{{\rm Y}},{\mathsf{\varpi }}_2\in \mathsf{\zeta })$ is a total subset of $\mathtt{P}$; its closed linear span is equal to $\mathtt{P}$;
• $\left(\mathsf{\vartheta }\left({\mathsf{\eta }}_1,{\mathsf{\eta }}_2\right)\mid \mathsf{\vartheta }\left({\mathsf{\eta }}_3,{\mathsf{\eta }}_4\right)\right)=\left({\mathsf{\eta }}_1\mid {\mathsf{\eta }}_2\right)\left({\mathsf{\eta }}_3\mid {\mathsf{\eta }}_4\right)$ for ${\mathsf{\eta }}_1,{\mathsf{\eta }}_2\in \mathsf{{\rm Y}},{\mathsf{\eta }}_3,{\mathsf{\eta }}_4\in \mathsf{\zeta }$. If $(\mathtt{P},\mathsf{\vartheta })$ is a tensor product of $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$, it is common to write ${\mathsf{\varpi }}_1\otimes {\mathsf{\varpi }}_2$ instead of $\mathsf{\vartheta }({\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2)$ and $\mathsf{{\rm Y}}\otimes \mathsf{\zeta }$ in place of $\mathtt{P}$. A tensor product of $\mathsf{{\rm Y}}$ with $\mathsf{\zeta }$ is a Hilbert space $\mathsf{{\rm Y}}\otimes \mathsf{\zeta }$ and a mapping $({\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2)\mapsto {\mathsf{\varpi }}_1\otimes {\mathsf{\varpi }}_2$ of $\mathsf{{\rm Y}}\times \mathsf{\zeta }$ goes into $\mathtt{G}\otimes \mathsf{\zeta }$ such that

  $$\begin{array}{cc}\hfill \left({\mathsf{\eta }}_1+{\mathsf{\eta }}_2\right)\otimes {\mathsf{\varpi }}_2& ={\mathsf{\eta }}_1\otimes {\mathsf{\varpi }}_2+{\mathsf{\eta }}_2\otimes {\mathsf{\varpi }}_2\hfill \\ \hfill \left(\mathsf{\lambda }{\mathsf{\varpi }}_1\right)\otimes {\mathsf{\varpi }}_2& =\mathsf{\lambda }({\mathsf{\varpi }}_1\otimes {\mathsf{\varpi }}_2)\hfill \\ \hfill {\mathsf{\varpi }}_1\otimes \left({\mathsf{\eta }}_3+{\mathsf{\eta }}_4\right)& ={\mathsf{\varpi }}_1\otimes {\mathsf{\eta }}_3+{\mathsf{\varpi }}_1\otimes {\mathsf{\eta }}_3\hfill \\ \hfill {\mathsf{\varpi }}_1\otimes \left(\lambda {\mathsf{\varpi }}_2\right)& =\mathsf{\lambda }({\mathsf{\varpi }}_1\otimes {\mathsf{\varpi }}_2).\hfill \end{array}$$

Let $\mathsf{\vartheta }:{\mathsf{\Omega }}_1\times \dots \times {\mathsf{\Omega }}_{\mathtt{s}}\to \mathsf{R}$ be a bounded real-valued mapping defined on the Cartesian product of the intervals. Let $\mathtt{D}=\left({\mathtt{D}}_1,\dots ,{\mathtt{D}}_{\mathtt{s}}\right)$ be an $\mathtt{s}$-tuple of self-adjoint operators on Hilbert spaces ${\mathtt{E}}_1,\dots ,{\mathtt{E}}_{\mathtt{s}}$ such that the spectra of each ${\mathtt{E}}_{\mathtt{i}}$ lies in ${\mathsf{\Omega }}_{\mathtt{i}}$ for every $\mathtt{i}=1,\dots ,\mathtt{s}$. Such an $\mathtt{s}$-tuple is referred to as being in the $\mathsf{\vartheta }$ domain. If

$${\mathtt{S}}_{\mathtt{i}}={\int }_{{\mathsf{\Omega }}_{\mathtt{i}}}{\mathsf{\delta }}_{\mathtt{i}}{\mathtt{dE}}_{\mathtt{i}}\left({\mathsf{\delta }}_{\mathtt{i}}\right)$$

is the spectrum of ${\mathtt{S}}_{\mathtt{i}}$ for $\mathtt{i}=1,\dots ,\mathtt{s}$, by adhering to [57], we define

$$\mathsf{\vartheta }\left({\mathtt{S}}_1,\dots ,{\mathtt{S}}_{\mathtt{s}}\right):={\int }_{{\mathsf{\Omega }}_1}\dots {\int }_{{\mathsf{\Omega }}_{\mathtt{s}}}\mathsf{\vartheta }\left({\mathsf{\delta }}_1,\dots ,{\mathsf{\delta }}_1\right){\mathtt{dE}}_1\left({\mathsf{\delta }}_1\right)\otimes \dots \otimes {\mathtt{dE}}_{\mathtt{z}}\left({\mathsf{\delta }}_{\mathtt{s}}\right)$$

within the tensorial product ${\mathtt{E}}_1\otimes \dots \otimes {\mathtt{E}}_{\mathtt{s}}$, which behaves as a bounded self-adjoint operator.

If the Hilbert spaces have finite dimensions, integration processes can be condensed to finite summations, making functional calculus more easily applied to real-valued functions. This construction [57] extends Korányi’s [75] concept for functions of two variables. It has the characteristic that

$$\mathsf{\vartheta }\left({\mathtt{S}}_1,\dots ,{\mathtt{S}}_{\mathtt{s}}\right)={\mathsf{\vartheta }}_1\left({\mathtt{S}}_1\right)\otimes \dots \otimes {\mathsf{\vartheta }}_{\mathtt{s}}\left({\mathtt{S}}_{\mathtt{s}}\right),$$

whenever $\mathsf{\vartheta }$ can be split as a product $\mathsf{\vartheta }\left({\mathtt{a}}_1,\dots ,{\mathtt{a}}_{\mathtt{s}}\right)={\mathsf{\vartheta }}_1\left({\mathtt{a}}_1\right)\dots {\mathsf{\vartheta }}_{\mathtt{s}}\left({\mathtt{a}}_{\mathtt{s}}\right)$ of $\mathtt{s}$ mappings each relying on just one variable.

On the interval $[0,\infty$), if $\mathsf{\vartheta }$ is super (sub)-multiplicative, then

$$\mathsf{\vartheta }\left({\mathsf{\varpi }}_1{\mathsf{\varpi }}_2\right)\ge (\le )\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)\mathrm{for}\mathrm{all}{\mathsf{\varpi }}_1{\mathsf{\varpi }}_2\in [0,\infty )$$

and if $\mathsf{\vartheta }$ is continuous on $[0,\infty )$, then

$$\mathsf{\vartheta }(\mathsf{{\rm Y}}\otimes \mathsf{\zeta })\ge (\le )\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)\otimes \mathsf{\vartheta }\left(\mathsf{\zeta }\right)\mathrm{for}\mathrm{all}\mathsf{{\rm Y}},\mathsf{\zeta }\ge 0.$$

This leads to the conclusion that if

$$\mathsf{{\rm Y}}={\int }_{[0,\infty )}{\mathsf{\varpi }}_1\mathtt{dE}\left({\mathsf{\varpi }}_1\right)\mathrm{and}\mathsf{\zeta }={\int }_{[0,\infty )}{\mathsf{\varpi }}_2\mathtt{dF}\left({\mathsf{\varpi }}_2\right)$$

are the spectral resolutions of $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$, then

$$\mathsf{\vartheta }(\mathsf{{\rm Y}}\otimes \mathsf{\zeta })={\int }_{[0,\infty )}{\int }_{[0,\infty )}\mathsf{\vartheta }\left({\mathsf{\varpi }}_1{\mathsf{\varpi }}_2\right)\mathtt{dE}\left({\mathsf{\varpi }}_1\right)\otimes \mathtt{dF}\left({\mathsf{\varpi }}_2\right)$$

for the $\mathsf{\vartheta }$ continuous function on $[0,\infty )$.

Where the geometric mean for positive operators $\mathsf{{\rm Y}},\mathsf{\zeta }>0$ is defined as follows

$$\mathsf{{\rm Y}}{\#}_{\mathtt{p}}\mathsf{\zeta }:={\mathsf{{\rm Y}}}^{1/2}{\left({\mathsf{{\rm Y}}}^{-1/2}\mathsf{\zeta }{\mathsf{{\rm Y}}}^{-1/2}\right)}^{\mathtt{p}}{\mathsf{{\rm Y}}}^{1/2},$$

where $\mathtt{p}\in [0,1]$ and

$$\mathsf{{\rm Y}}\#\mathsf{\zeta }:={\mathsf{{\rm Y}}}^{1/2}{\left({\mathsf{{\rm Y}}}^{-1/2}\mathsf{\zeta }{\mathsf{{\rm Y}}}^{-1/2}\right)}^{1/2}{\mathsf{{\rm Y}}}^{1/2}.$$

By the definitions of # and ⊗, we have

$$\mathsf{{\rm Y}}\#\mathsf{\zeta }=\mathsf{\zeta }\#\mathsf{{\rm Y}}\mathrm{and}(\mathsf{{\rm Y}}\#\mathsf{\zeta })\otimes (\mathsf{\zeta }\#\mathsf{{\rm Y}})=(\mathsf{{\rm Y}}\otimes \mathsf{\zeta })\#(\mathsf{\zeta }\otimes \mathsf{{\rm Y}}).$$

Consider the subsequent characteristic of the tensorial product:

$$\left(\mathsf{{\rm Y}}\mathtt{J}\right)\otimes \left(\mathsf{\zeta }\mathtt{L}\right)=(\mathsf{{\rm Y}}\otimes \mathsf{\zeta })(\mathtt{J}\otimes \mathtt{L})$$

which holds for all $\mathsf{{\rm Y}},\mathsf{\zeta },\mathtt{J},\mathtt{L}\in \mathtt{B}\left({\mathsf{\varpi }}_{\mathtt{2}}\right)$. If we take $\mathtt{J}=\mathsf{{\rm Y}}$ and $\mathtt{L}=\mathsf{\zeta }$, then we obtain

$${\mathsf{{\rm Y}}}^2\otimes {\mathsf{\zeta }}^2={(\mathsf{{\rm Y}}\otimes \mathsf{\zeta })}^2.$$

Through induction, we have

$${\mathsf{{\rm Y}}}^{\mathtt{s}}\otimes {\mathsf{\zeta }}^{\mathtt{s}}={(\mathsf{{\rm Y}}\otimes \mathsf{\zeta })}^{\mathtt{s}}\mathrm{for}\mathrm{natural}\mathtt{s}\ge 0.$$

In particular,

$${\mathsf{{\rm Y}}}^{\mathtt{s}}\otimes 1={(\mathsf{{\rm Y}}\otimes 1)}^{\mathtt{s}}\mathrm{and}1\otimes {\mathsf{\zeta }}^{\mathtt{s}}={(1\otimes \mathsf{\zeta })}^{\mathtt{s}}$$

for all $\mathtt{s}\ge 0$. Additionally, we note that the operations $1\otimes \mathsf{\zeta }$ and $\mathsf{{\rm Y}}\otimes 1$ are commutative and

$$(\mathsf{{\rm Y}}\otimes 1)(1\otimes \mathsf{\zeta })=(1\otimes \mathsf{\zeta })(\mathsf{{\rm Y}}\otimes 1)=\mathsf{{\rm Y}}\otimes \mathsf{\zeta }.$$

Moreover, we possess for two natural numbers $\mathtt{s},\mathtt{t}$

$${(\mathsf{{\rm Y}}\otimes 1)}^{\mathtt{s}}{(1\otimes \mathsf{\zeta })}^{\mathtt{t}}={(1\otimes \mathsf{\zeta })}^{\mathtt{s}}{(\mathsf{{\rm Y}}\otimes 1)}^{\mathtt{t}}={\mathsf{{\rm Y}}}^{\mathtt{t}}\otimes {\mathsf{\zeta }}^{\mathtt{s}}.$$

Variable Exponent Spaces

This subsection provides some definitions and related results that are necessary to proceed with the article, specifically for developing the double Hermite–Hadamard inequality in Section 6.

Definition 3 

(see [71]). We consider variable exponents functions on ${\mathsf{R}}^{\mathtt{n}}$ whose range is in $\left[\mathtt{m},\infty \right]$ for some $\mathtt{m}>0$.

We define some notation to describe the range of exponent functions. We use the standard notation

$${\mathtt{p}}^{+}=\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}}}{ess}sup\mathtt{p}\left(\sigma \right),{\mathtt{p}}^{-}=\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}}}{ess\; inf}\mathtt{p}\left(\sigma \right).$$

Some examples of exponent functions on $\mathsf{R}$ include $\mathtt{p}\left(\sigma \right)=\mathtt{p}$ for some constant $\mathtt{p}$, or $\mathtt{p}\left(\sigma \right)=2+sin\left(\sigma \right)$.

Definition 4 

(see [71]). The variable exponent Lebesgue space is ${\mathtt{L}}_{\mathtt{p}}(·)$ and $\left({\mathsf{R}}^{\mathtt{n}}\right)$ is the set of all measurable functions $\mathsf{\vartheta }$ on ${\mathsf{R}}^{\mathtt{n}}$ that have the modular

$${\mathsf{\rho }}_{\mathtt{p}(·)}(\mathsf{\vartheta }/\mathsf{\lambda })={\int }_{{\mathsf{R}}^{\mathtt{n}}}{\left|{\displaystyle \frac{\mathsf{\vartheta }\left(\sigma \right)}{\mathsf{\lambda }}}\right|}^{\mathtt{p}\left(\sigma \right)}\mathtt{d}\sigma$$

which is finite for some $0<\mathsf{\lambda }$. This is a Banach function space related to the following quasi-norm:

$${\parallel \mathsf{\vartheta }\parallel }_{\mathtt{p}(·)}=inf\left\{\mathsf{\lambda }>0:{\mathsf{\rho }}_{\mathtt{p}(·)}(\mathsf{\vartheta }/\mathsf{\lambda })⩽1\right\}.$$

This functional defines a norm when ${\mathtt{p}}^{-}⩾1$.

Remark 1. 

Let $\mathsf{\Omega }\subset {\mathsf{R}}^{\mathtt{n}}$ be open-set. We set

$${\mathtt{L}}_{\mathtt{p}}(·)\left(\mathsf{\Omega }\right)=\left\{\mathsf{\vartheta }:\mathsf{\Omega }\to \mathsf{C}\mathit{mesurable}:\mathsf{\vartheta }{\mathsf{\chi }}_{\mathsf{\Omega }}\in {\mathtt{L}}_{\mathtt{p}}(·)\left({\mathsf{R}}^{\mathtt{n}}\right)\right\}$$

and

$${\parallel \mathsf{\vartheta }\parallel }_{{\mathtt{L}}^{\mathtt{p}(·)}\left(\mathsf{\Omega }\right)}={∥\mathsf{\vartheta }{\mathsf{\chi }}_{\mathsf{\Omega }}∥}_{{\mathtt{L}}^{\mathtt{p}(·)}\left({\mathsf{R}}^{\mathtt{n}}\right)}.$$

Example 1. 

Suppose that $\mathtt{p}\left(\sigma \right)=2{\mathsf{\chi }}_{[0,2]}+\left({\sigma }^2+1\right){\mathsf{\chi }}_{\mathsf{R}\setminus [0,2]}$ and consider the function $\mathsf{\vartheta }\left(\sigma \right)=$ $\sqrt{2\sigma }{e}^{{\sigma }^2/2}{\mathsf{\chi }}_{[0,2]}$. Then,

$${\mathsf{\rho }}_{\mathtt{p}(·)}(\mathsf{\vartheta }/\mathsf{\lambda })={\int }_{\mathsf{R}}{\left|{\displaystyle \frac{\sqrt{2\sigma }{e}^{{\sigma }^2/2}}{\mathsf{\lambda }}}\right|}^{\mathtt{p}\left(\sigma \right)}\mathtt{d}\sigma ={\displaystyle \frac{1}{{\mathsf{\lambda }}^2}}{\int }_0^{2}2\sigma e^{{\sigma }^2}\mathtt{d}\sigma ={\displaystyle \frac{{\mathtt{e}}^4-1}{{\mathsf{\lambda }}^2}}.$$

Therefore, ${\parallel \mathsf{\vartheta }\parallel }_{\mathtt{p}(·)}=\sqrt{{\mathtt{e}}^4-1}$, and then $\mathsf{\vartheta }\in {\mathtt{L}}_{\mathtt{p}}(·)\left(\mathsf{R}\right)$.

In a similar spirit, one can define variable exponent sequence spaces ${\ell }_{\mathtt{p}(·)}$, where the exponent $\mathtt{p}$ varies with the index of the sequence:

$${\ell }_{\mathtt{p}(·)}=\left\{\left\{{\xi }_{\mathsf{\eta }}\right\}\subset \mathsf{R};\sum _{\mathsf{\eta }=0}^{\infty }{\left|\mathsf{\lambda }{\xi }_{\mathsf{\eta }}\right|}^{\mathtt{p}\left(\mathsf{\eta }\right)}<\infty \mathrm{for}\mathrm{some}\mathsf{\lambda }>0\right\},$$

where $\left\{\mathtt{p}\right(\mathsf{\eta }\left)\right\}\subset [1,\infty )$. A comprehensive assessment of these spaces may be found in [76,77]. The sequence vector space is now defined as follows:

$${\ell }_{\mathtt{p}(·)}=\left\{\left\{{\xi }_{\mathsf{\eta }}\right\}\subset \mathsf{R};\sum _{\mathsf{\eta }=0}^{\infty }{\displaystyle \frac{1}{\mathtt{p}\left(\nu \right)}}{\left|\mathsf{\lambda }{\xi }_{\mathsf{\eta }}\right|}^{\mathtt{p}\left(\mathsf{\eta }\right)}<\infty \mathrm{for}\mathrm{some}\mathsf{\lambda }>0\right\},$$

if $\mathtt{p}:\mathsf{\Omega }\to [1,\infty ).$

Modular vector spaces [77] build on the concept of normed vector spaces by employing a modular function, allowing for a wider range of applications and more generic frameworks in mathematical analysis. Let $\mathtt{v}:{\ell }_{\mathtt{p}(·)}\to [0,\infty ]$ be defined as

$$\mathtt{v}\left(\mathsf{\eta }\right)=\mathtt{v}\left(\left({\xi }_{\mathsf{\eta }}\right)\right)=\sum _{\mathsf{\eta }=0}^{\infty }{\displaystyle \frac{1}{\mathtt{p}\left(\mathsf{\eta }\right)}}{\left|{\xi }_{\mathsf{\eta }}\right|}^{\mathtt{p}\left(\mathsf{\eta }\right)}.$$

Then, the following properties apply:

• $\mathtt{v}\left(\mathsf{\eta }\right)=0$ iff $\mathsf{\eta }=0$;
• $\mathtt{v}\left(\mathsf{\eta }\mathsf{\eta }\right)=\mathtt{v}\left(\mathsf{\eta }\right)$, if $\left|\mathsf{\eta }\right|=1$;
• $\mathtt{v}(\mathsf{\sigma }{\mathsf{\eta }}_1+(1-\mathsf{\sigma }){\mathsf{\eta }}_2)\le \mathsf{\sigma }\mathtt{v}\left({\mathsf{\eta }}_1\right)+(1-\mathsf{\sigma })\mathtt{v}\left({\mathsf{\eta }}_2\right)$ for any $\mathsf{\sigma }\in [0,1]$.

for any ${\mathsf{\eta }}_1,{\mathsf{\eta }}_2\in {\ell }_{\mathtt{p}(·)}$.

Definition 5 

(see [78]). Let $\mathtt{p},\mathtt{q}\in \mathcal{P}\left({\mathsf{R}}^{\mathtt{n}}\right)$. The mixed Lebesgue-sequence space ${\ell }_{\mathtt{q}(·)}\left({\mathtt{L}}_{\mathtt{p}}(·)\right)$ is defined on sequences of ${\mathtt{L}}_{\mathtt{p}}(·)$ and defined as follows:

$${\varrho }_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{L}}_{\mathtt{p}}(·)\right.}\left({\xi }_{\mathsf{\eta }}\right)=\sum _{\mathsf{\eta }=0}^{\infty }inf\left\{{\varsigma }_{\mathsf{\eta }}>0:{\varrho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\xi }_{\mathsf{\eta }}}{{\varsigma }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}.$$

The (quasi)-norm is defined as follows:

$${∥{\xi }_{\mathsf{\eta }}∥}_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{L}}_{\mathtt{p}}(·)\right)}=inf\left\{\mathsf{\mu }>0:{\varrho }_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{L}}_{\mathtt{p}}(·)\right)}\left({\displaystyle \frac{{\xi }_{\mathsf{\eta }}}{\mathsf{\mu }}}\right)\le 1\right\}.$$

Furthermore, if $\mathtt{p}$ and $\mathtt{q}$ are constants, then ${\ell }_{\mathtt{q}(·)}\left({\mathtt{L}}_{\mathtt{p}}(·)\right)={\ell }_{\mathtt{q}}\left({\mathtt{L}}_{\mathtt{p}}\right)$.

Example 2. 

Let $\mathtt{p}\equiv \infty ,\left\{{\mathsf{\varphi }}_{\mathsf{\eta }}\right\}$ be a sequence as follows: ${\mathsf{\varphi }}_{\mathsf{\eta }}\left(\sigma \right)={\displaystyle \frac{e^{-{\sigma }^2}}{v}}{\mathsf{\chi }}_{[-1;\infty )}$ for all $v\in \mathsf{N}$, and consider the function $\mathtt{q}\left(\sigma \right)=2{\mathsf{\chi }}_{[-1;1]}+3{\mathsf{\chi }}_{\mathsf{R}\setminus [-1;1]}$. Then,

$${\mathsf{\rho }}_{{\ell }^{\mathtt{q}(·)}\left({\mathtt{L}}^{\infty }\right)}\left(\left({\mathsf{\varphi }}_{\mathsf{\eta }}\right)\right)=\sum _{v}inf\left\{{\mathsf{\eta }}_v>0:{\mathsf{\rho }}_{\infty }\left({\mathsf{\varphi }}_{\mathsf{\eta }}/{\mathsf{\eta }}_v^{1/\mathtt{q}(·)}\right)⩽1\right\}.$$

Now, ${\mathsf{\rho }}_{\infty }\left(\eta \right)⩽1$ iff $\left|\eta \right|⩽1$ almost everywhere. Thus, $\left|{\mathsf{\varphi }}_{\mathsf{\eta }}\right|/\left({\mathsf{\eta }}_v^{1/\mathtt{q}(·)}\right)⩽1$ and hence ${\mathsf{\eta }}_v⩾{esssup}_{\sigma }{\left|{\mathsf{\varphi }}_{\mathsf{\eta }}\left(\sigma \right)\right|}^{\mathtt{q}\left(\sigma \right)}$. It follows that

$${\mathsf{\rho }}_{\ell \mathtt{q}(·)(\mathtt{L}\infty )}\left({\displaystyle \frac{1}{\mu }}\left({\mathsf{\varphi }}_{\mathsf{\eta }}\right)\right)=\sum _{v}ess\underset{\sigma }{sup}{\left|{\displaystyle \frac{{\mathsf{\varphi }}_{\mathsf{\eta }}\left(\sigma \right)}{\mu }}\right|}^{\mathtt{q}\left(\sigma \right)}=\sum _{v}ess\underset{-1⩽\sigma ⩽1}{sup}{\displaystyle \frac{e^{-2{\sigma }^2}}{v^2{\mu }^2}}={\displaystyle \frac{{\pi }^2}{6{\mu }^2}},$$

and so ${∥{\left({\mathsf{\varphi }}_{\mathsf{\eta }}\right)}_v∥}_{{\ell }^{\mathtt{q}(·)}\left({\mathtt{L}}^{\infty }\left(\mathsf{R}\right)\right)}={\displaystyle \frac{\pi }{\sqrt{6}}}$, then ${\mathsf{\varphi }}_{\mathsf{\eta }}\in {\ell }^{\mathtt{q}(·)}\left({\mathtt{L}}^{\infty }\left(\mathsf{R}\right)\right)$.

Now we recall a main definition related to mixed Morrey-sequence space for variable exponents that we utilize in proving our main findings.

Definition 6 

(see [79]). Let $\mathtt{p}(·),\mathtt{q}(·),\mathtt{v}(·)\in \mathcal{P}\left({\mathsf{R}}^{\mathtt{n}}\right)$ with $\mathtt{p}\left(\sigma \right)\le \mathtt{v}\left(\sigma \right)$. The mixed Morrey-sequence space ${\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}(·),\mathtt{v}(·)}\right)$ consists of all sequences ${\left({\mathsf{\varphi }}_{\mathsf{\eta }}\right)}_{\nu }\subset \mathcal{M}\left({\mathsf{R}}^{\mathtt{n}}\right)$ such that

$${\mathsf{\rho }}_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}(·),\mathtt{v}(·)}\right)}\left({\left({\mathsf{\varphi }}_{\mathsf{\eta }}\right)}_{\nu }\right):=\sum _{\nu \ge 0}\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}inf\left\{{\mathsf{\eta }}_{\nu }>0:{\mathsf{\rho }}_{\mathtt{p}(·)}\left({\displaystyle \frac{{\mathtt{r}}^{\frac{n}{\mathtt{v}\left(\sigma \right)}-\frac{n}{\mathtt{p}\left(\sigma \right)}}{\mathsf{\varphi }}_{\nu }{\chi }_{\mathtt{B}(\sigma ,\mathtt{r})}}{{\mathsf{\eta }}_{\nu }^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}.$$

Definition 7 

(see [80]). Let $\mathsf{\vartheta }:[{\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2]\to \mathsf{R}$ be a Lebesgue integrable continuous function on $[{\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2]$. The $\mathtt{RLF}$ integrals are defined for $\mathsf{\eta }>0$ by

$${\mathsf{J}}_{{\mathsf{\varpi }}_1+}^{\mathsf{\eta }}\mathsf{\vartheta }\left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\eta }\right)}}{\int }_{{\mathsf{\varpi }}_1}^x{(x-\mathsf{\eta })}^{\mathsf{\eta }-1}\mathsf{\vartheta }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }$$

for ${\mathsf{\varpi }}_1<x⩽{\mathsf{\varpi }}_2$ and

$${\mathsf{J}}_{{\mathsf{\varpi }}_2-}^{\mathsf{\eta }}\mathsf{\vartheta }\left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\eta }\right)}}{\int }_{\mathtt{x}}^{{\mathsf{\varpi }}_2}{(\mathsf{\eta }-x)}^{\mathsf{\eta }-1}\mathsf{\vartheta }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }$$

for ${\mathsf{\varpi }}_1⩽x<{\mathsf{\varpi }}_2$, where Γ is the special function.

3. The Main Results

The following representation results for continuous functions, obtained by using several arithmetic operations, serve as the foundation for our primary conclusions:

Lemma 1. 

Let $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ be self-adjoint operators with $\mathtt{Sp}\left(\mathsf{{\rm Y}}\right)\subset {\mathsf{\Omega }}_1$ and $\mathtt{Sp}\left(\mathsf{\zeta }\right)\subset$ ${\mathsf{\Omega }}_2$. Assume $\mathsf{\vartheta },\mathsf{\psi }$ are continuous on ${\mathsf{\Omega }}_1$, and Φ, $\mathsf{\vartheta }$ continuous on ${\mathsf{\Omega }}_2$ and $\mathsf{\phi }$ continuous on Ω; this comprises the sum of the intervals $\mathsf{\psi }\left({\mathsf{\Omega }}_1\right)+\mathsf{\vartheta }\left({\mathsf{\Omega }}_2\right)$, and then one has

$$\begin{array}{cc}\hfill & \left(\mathsf{\vartheta }\right(\mathsf{{\rm Y}})\otimes 1+1\otimes \mathsf{\Phi }(\mathsf{\zeta }\left)\right)\mathsf{\phi }\left(\mathsf{\psi }\right(\mathsf{{\rm Y}})\otimes 1+1\otimes \mathsf{\vartheta }(\mathsf{\zeta }\left)\right)\hfill \\ \hfill & ={\int }_{{\mathsf{\Omega }}_1}{\int }_{{\mathsf{\Omega }}_2}(\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)+\mathsf{\Phi }\left({\mathsf{\varpi }}_1\right))\mathsf{\phi }(\mathsf{\psi }\left({\mathsf{\varpi }}_2\right)+\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right))\mathtt{d}{\mathtt{E}}_{{\mathsf{\Omega }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\Omega }}_2},\hfill \end{array}$$

(1)

where $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ have the spectral resolutions

$$\mathsf{{\rm Y}}={\int }_{{\mathsf{\Omega }}_1}{\mathsf{\varpi }}_2\mathtt{dE}\left({\mathsf{\varpi }}_2\right)and\mathsf{\zeta }={\int }_{{\mathsf{\Omega }}_2}{\mathsf{\varpi }}_1\mathtt{dF}\left({\mathsf{\varpi }}_1\right).$$

Proof. 

Stone–Weierstrass state that a polynomial sequence can be used to approximate any continuous function; thus, verifying the equality of the polynomial function is sufficient. Consider that $\mathsf{\phi }\left(\mathsf{\mu }\right)={\mathsf{\mu }}^{\mathtt{n}}$ with $\mathtt{n}$ is any natural number, and then we have

$$\begin{array}{cc}\hfill \mathcal{G}& :={\int }_{{\mathsf{\Omega }}_1}{\int }_{{\mathsf{\Omega }}_2}(\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)+\mathsf{\Phi }\left({\mathsf{\varpi }}_1\right)){(\mathsf{\psi }\left({\mathsf{\varpi }}_2\right)+\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right))}^{\mathtt{n}}\mathtt{d}{\mathtt{E}}_{{\mathsf{\Omega }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\Omega }}_2}\hfill \\ & ={\int }_{{\mathsf{\Omega }}_1}{\int }_{{\mathsf{\Omega }}_2}(\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)+\mathsf{\Phi }\left({\mathsf{\varpi }}_1\right))\sum _{\mathtt{m}=\mathtt{0}}^{\mathtt{n}}{C}_{\mathtt{n}}^{\mathtt{m}}{\left[\mathsf{\psi }\left({\mathsf{\varpi }}_2\right)\right]}^{\mathtt{m}}{\left[\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)\right]}^{\mathtt{n}-\mathtt{m}}\mathtt{d}{\mathtt{E}}_{{\mathsf{\Omega }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\Omega }}_2}\hfill \\ & =\sum _{\mathtt{m}=\mathtt{0}}^{\mathtt{n}}{C}_{\mathtt{n}}^{\mathtt{m}}{\int }_{{\mathsf{\Omega }}_1}{\int }_{{\mathsf{\Omega }}_2}(\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)+\mathsf{\Phi }\left({\mathsf{\varpi }}_1\right)){\left[\mathsf{\psi }\left({\mathsf{\varpi }}_2\right)\right]}^{\mathtt{m}}{\left[\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)\right]}^{\mathtt{n}-\mathtt{m}}\mathtt{d}{\mathtt{E}}_{{\mathsf{\Omega }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\Omega }}_2}\hfill \\ & =\sum _{\mathtt{m}=\mathtt{0}}^{\mathtt{n}}{C}_{\mathtt{n}}^{\mathtt{m}}\left[{\int }_{{\mathsf{\Omega }}_1}{\int }_{{\mathsf{\Omega }}_2}\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right){\left[\mathsf{\psi }\left({\mathsf{\varpi }}_2\right)\right]}^{\mathtt{m}}{\left[\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)\right]}^{\mathtt{n}-\mathtt{m}}\mathtt{d}{\mathtt{E}}_{{\mathsf{\Omega }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\Omega }}_2}\right.\hfill \\ & \left.+{\int }_{{\mathsf{\Omega }}_1}{\int }_{{\mathsf{\Omega }}_2}{\left[\mathsf{\psi }\left({\mathsf{\varpi }}_2\right)\right]}^{\mathtt{m}}\mathsf{\Phi }\left({\mathsf{\varpi }}_1\right){\left[\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)\right]}^{\mathtt{n}-\mathtt{m}}\mathtt{d}{\mathtt{E}}_{{\mathsf{\Omega }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\Omega }}_2}\right].\hfill \end{array}$$

Observe that

$$\begin{array}{cc}& {\int }_{{\mathsf{\Omega }}_1}{\int }_{{\mathsf{\Omega }}_2}\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right){\left[\mathsf{\psi }\left({\mathsf{\varpi }}_2\right)\right]}^{\mathtt{m}}{\left[\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)\right]}^{\mathtt{n}-\mathtt{m}}\mathtt{d}{\mathtt{E}}_{{\mathsf{\Omega }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\Omega }}_2}\hfill \\ & =\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right){\left[\mathsf{\psi }\left(\mathsf{{\rm Y}}\right)\right]}^{\mathtt{m}}\otimes {\left[\mathsf{\vartheta }\left(\mathsf{\zeta }\right)\right]}^{\mathtt{n}-\mathtt{m}}=(\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)\otimes 1)\left({\left[\mathsf{\psi }\left(\mathsf{{\rm Y}}\right)\right]}^{\mathtt{m}}\otimes {\left[\mathsf{\vartheta }\left(\mathsf{\zeta }\right)\right]}^{\mathtt{n}-\mathtt{m}}\right)\hfill \\ & =(\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)\otimes 1)\left({\left[\mathsf{\psi }\left(\mathsf{{\rm Y}}\right)\right]}^{\mathtt{m}}\otimes 1\right)\left(1\otimes {\left[\mathsf{\vartheta }\left(\mathsf{\zeta }\right)\right]}^{\mathtt{n}-\mathtt{m}}\right)\hfill \\ & =(\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)\otimes 1){(\mathsf{\psi }\left(\mathsf{{\rm Y}}\right)\otimes 1)}^{\mathtt{m}}{(1\otimes \mathsf{\vartheta }\left(\mathsf{\zeta }\right))}^{\mathtt{n}-\mathtt{m}}\hfill \end{array}$$

and

$$\begin{array}{cc}& {\int }_{{\mathsf{\Omega }}_1}{\int }_{{\mathsf{\Omega }}_2}{\left[\mathsf{\psi }\left({\mathsf{\varpi }}_2\right)\right]}^{\mathtt{m}}\mathsf{\Phi }\left({\mathsf{\varpi }}_1\right){\left[\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)\right]}^{\mathtt{n}-\mathtt{m}}\mathtt{d}{\mathtt{E}}_{{\mathsf{\Omega }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\Omega }}_2}\hfill \\ & ={\left[\mathsf{\psi }\left(\mathsf{{\rm Y}}\right)\right]}^{\mathtt{m}}\otimes \left(\mathsf{\Phi }\left(\mathsf{\zeta }\right){\left[\mathsf{\vartheta }\left(\mathsf{\zeta }\right)\right]}^{\mathtt{n}-\mathtt{m}}\right)=(1\otimes \mathsf{\Phi }\left(\mathsf{\zeta }\right))\left({\left[\mathsf{\psi }\left(\mathsf{{\rm Y}}\right)\right]}^{\mathtt{m}}\otimes {\left[\mathsf{\vartheta }\left(\mathsf{\zeta }\right)\right]}^{\mathtt{n}-\mathtt{m}}\right)\hfill \\ & =(1\otimes \mathsf{\Phi }\left(\mathsf{\zeta }\right))\left({\left[\mathsf{\psi }\left(\mathsf{{\rm Y}}\right)\right]}^{\mathtt{m}}\otimes 1\right)\left(1\otimes {\left[\mathsf{\vartheta }\left(\mathsf{\zeta }\right)\right]}^{\mathtt{n}-\mathtt{m}}\right)\hfill \\ & =(1\otimes \mathsf{\Phi }\left(\mathsf{\zeta }\right)){(\mathsf{\psi }\left(\mathsf{{\rm Y}}\right)\otimes 1)}^{\mathtt{m}}{(1\otimes \mathsf{\vartheta }\left(\mathsf{\zeta }\right))}^{\mathtt{n}-\mathtt{m}}\hfill \end{array}$$

with $\left(\mathsf{\psi }\right(\mathsf{{\rm Y}})\otimes 1)$ and $(1\otimes \mathsf{\vartheta }(\mathsf{\zeta }\left)\right)$ commutative. Therefore,

$$\begin{array}{cc}\hfill \mathcal{G}& =(\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)\otimes 1+1\otimes \mathsf{\Phi }\left(\mathsf{\zeta }\right))\sum _{\mathtt{m}=\mathtt{0}}^{\mathtt{n}}{C}_{\mathtt{n}}^{\mathtt{m}{\left(\mathsf{\psi }\right(\mathsf{{\rm Y}})\otimes 1)}^{\mathtt{m}}{(1\otimes \mathsf{\vartheta }\left(\mathsf{\zeta }\right))}^{\mathtt{n}-\mathtt{m}}}\hfill \\ & =(\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)\otimes 1+1\otimes \mathsf{\Phi }\left(\mathsf{\zeta }\right)){(\mathsf{\psi }\left(\mathsf{{\rm Y}}\right)\otimes 1+1\otimes \mathsf{\vartheta }\left(\mathsf{\zeta }\right))}^{\mathtt{n}},\hfill \end{array}$$

whereby the commutativity of $1\otimes \mathsf{\vartheta }\left(\mathsf{\zeta }\right)$ and $\mathsf{\psi }\left(\mathsf{{\rm Y}}\right)\otimes 1$ was used. □

Lemma 2. 

Let $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ be self-adjoint operators with $\mathtt{Sp}\left(\mathsf{{\rm Y}}\right)\subset {\mathsf{\Omega }}_1$ and $\mathtt{Sp}\left(\mathsf{\zeta }\right)\subset$ ${\mathsf{\Omega }}_2$. Assume $\mathsf{\vartheta },\mathsf{\psi }$ are continuous on ${\mathsf{\Omega }}_1$, and $\mathsf{\Phi },\mathsf{\vartheta }$ continuous on ${\mathsf{\Omega }}_2$ and $\mathsf{\phi }$ continuous on Ω; this comprises the product of the intervals $\mathsf{\vartheta }\left({\mathsf{\Omega }}_1\right)\mathsf{\Phi }\left({\mathsf{\Omega }}_2\right),\mathsf{\vartheta }\left({\mathsf{\Omega }}_1\right)\mathsf{\vartheta }\left({\mathsf{\Omega }}_2\right)$, and then one has

$$\begin{array}{c}\hfill \mathsf{\phi }(\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)\otimes \mathsf{\Phi }\left(\mathsf{\zeta }\right))\mathsf{\chi }(\mathsf{\psi }\left(\mathsf{{\rm Y}}\right)\otimes \mathsf{\vartheta }\left(\mathsf{\zeta }\right))={\int }_{{\mathsf{\Omega }}_1}{\int }_{{\mathsf{\Omega }}_2}\mathsf{\phi }\left(\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)\mathsf{\Phi }\left({\mathsf{\varpi }}_1\right)\right)\mathsf{\chi }\left(\mathsf{\psi }\left({\mathsf{\varpi }}_2\right)\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)\right)\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\varpi }}_2}\end{array}$$

(2)

where $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ have the spectral resolutions

$$\mathsf{{\rm Y}}={\int }_{{\mathsf{\Omega }}_1}{\mathsf{\varpi }}_2\mathtt{dE}\left({\mathsf{\varpi }}_2\right)and\mathsf{\zeta }={\int }_{{\mathsf{\Omega }}_2}{\mathsf{\varpi }}_1\mathtt{dF}\left({\mathsf{\varpi }}_1\right).$$

Proof. 

Stone–Weierstrass state that a polynomial sequence can be used to approximate any continuous function; thus, verifying the equality of the polynomial function is sufficient. Consider $\mathsf{\phi }\left(\mathsf{\mu }\right)={\mathsf{\mu }}^{\mathtt{m}}$ and $\mathsf{\chi }\left(\mathsf{\mu }\right)={\mathsf{\mu }}^{\mathtt{n}}$ with $\mathtt{n}$ and $\mathtt{m}$ any natural numbers. We have

$$\begin{array}{cc}& {\int }_{{\mathsf{\Omega }}_1}{\int }_{{\mathsf{\Omega }}_2}{\left({\mathsf{\varpi }}_1{\mathsf{\varpi }}_2\right)}^{\mathtt{m}}{\left({\mathsf{\varpi }}_1{\mathsf{\varpi }}_2\right)}^{\mathtt{n}}\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\varpi }}_2}\hfill \\ & ={\int }_{{\mathsf{\Omega }}_1}{\int }_{{\mathsf{\Omega }}_2}{\left[{\mathsf{\varpi }}_1\right]}^{\mathtt{m}}{\left[{\mathsf{\varpi }}_2\right]}^{\mathtt{m}}{\left[{\mathsf{\varpi }}_1\right]}^{\mathtt{n}}{\left[{\mathsf{\varpi }}_2\right]}^{\mathtt{n}}\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\varpi }}_2}\hfill \\ & ={\int }_{{\mathsf{\Omega }}_1}{\int }_{{\mathsf{\Omega }}_2}{\left[{\mathsf{\varpi }}_1\right]}^{\mathtt{m}}{\left[{\mathsf{\varpi }}_1\right]}^{\mathtt{n}}{\left[{\mathsf{\varpi }}_2\right]}^{\mathtt{m}}{\left[{\mathsf{\varpi }}_2\right]}^{\mathtt{n}}\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\varpi }}_2}\hfill \\ & =\left({\left[\mathsf{{\rm Y}}\right]}^{\mathtt{m}}{\left[\mathsf{{\rm Y}}\right]}^{\mathtt{n}}\right)\otimes \left({\left[\mathsf{\zeta }\right]}^{\mathtt{m}}{\left[\mathsf{\zeta }\right]}^{\mathtt{n}}\right)\hfill \\ & =\left({\left[\mathsf{{\rm Y}}\right]}^{\mathtt{m}}\otimes {\left[\mathsf{\zeta }\right]}^{\mathtt{m}}\right)\left({\left[\mathsf{{\rm Y}}\right]}^{\mathtt{n}}\otimes {\left[\mathsf{\zeta }\right]}^{\mathtt{n}}\right)\hfill \\ & ={(\mathsf{{\rm Y}}\otimes \mathsf{\zeta })}^{\mathtt{m}}{(\mathsf{{\rm Y}}\otimes \mathsf{\zeta })}^{\mathtt{n}}\hfill \end{array}$$

and the equality $\left(2\right)$ is proven. □

Some Needed Technical Identities for the Riemann–Liouville Fractional Integral Operators

In this part, we use the following formulation of the Riemann–Liouville fractional integral to establish various fractional identities that we employ in numerous major results.

Lemma 3. 

Let $\mathsf{\vartheta }:[{\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2]\to \mathsf{R}$ be an absolutely continuous function on $[{\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2]$.

• For any $\mathtt{x}\in ({\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2)$, we have

$$\begin{array}{cc}\hfill & {\mathsf{J}}_{{\mathsf{\varpi }}_1+}^{\mathsf{\eta }}\mathsf{\vartheta }\left(x\right)+{\mathsf{J}}_{{\mathsf{\varpi }}_2-}^{\mathsf{\eta }}\mathsf{\vartheta }\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}[{(x-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)+{({\mathsf{\varpi }}_2-\mathtt{x})}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{\zeta }\right)]\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left[{\int }_{{\mathsf{\varpi }}_1}^x{(x-\mathsf{\eta })}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }-{\int }_{\mathtt{x}}^{{\mathsf{\varpi }}_2}{(\mathsf{\eta }-\mathtt{x})}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }\right].\hfill \end{array}$$

(3)

Proof. 

Since $\mathsf{\vartheta }:[{\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2]\to \mathsf{R}$ is an absolutely continuous function on $[{\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2]$, then the integrals

$${\int }_{{\mathsf{\varpi }}_1}^x{(x-\mathsf{\eta })}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }\mathrm{and}{\int }_{\mathtt{x}}^{{\mathsf{\varpi }}_2}{(\mathsf{\eta }-\mathtt{x})}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }$$

exist, and integrating by parts, we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\int }_{{\mathsf{\varpi }}_1}^x{(x-\mathsf{\eta })}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }\hfill \\ \hfill & ={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\eta }\right)}}{\int }_{{\mathsf{\varpi }}_1}^x{(x-\mathsf{\eta })}^{\mathsf{\eta }-1}\mathsf{\vartheta }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }-{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{(x-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)\hfill \\ \hfill & ={\mathsf{J}}_{{\mathsf{\varpi }}_1+}^{\mathsf{\eta }}\mathsf{\vartheta }\left(x\right)-{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{(x-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)\hfill \end{array}$$

(4)

for ${\mathsf{\varpi }}_1<x⩽{\mathsf{\varpi }}_2$ and

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\int }_{\mathtt{x}}^{{\mathsf{\varpi }}_2}{(\mathsf{\eta }-\mathtt{x})}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }\hfill \\ \hfill & ={\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{({\mathsf{\varpi }}_2-\mathtt{x})}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{\zeta }\right)-{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\eta }\right)}}{\int }_{\mathtt{x}}^{{\mathsf{\varpi }}_2}{(\mathsf{\eta }-\mathtt{x})}^{\mathsf{\eta }-1}\mathsf{\vartheta }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }\hfill \\ \hfill & ={\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{({\mathsf{\varpi }}_2-\mathtt{x})}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{\zeta }\right)-{\mathsf{J}}_{{\mathsf{\varpi }}_2-}^{\mathsf{\eta }}\mathsf{\vartheta }\left(x\right)\hfill \end{array}$$

(5)

for ${\mathsf{\varpi }}_1⩽\mathtt{x}<{\mathsf{\varpi }}_2$. From (4), we have

$${\mathsf{J}}_{{\mathsf{\varpi }}_1+}^{\mathsf{\eta }}\mathsf{\vartheta }\left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{(x-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)+{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\int }_{{\mathsf{\varpi }}_1}^x{(x-\mathsf{\eta })}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }$$

for ${\mathsf{\varpi }}_1<x⩽{\mathsf{\varpi }}_2$, and from (5), we have

$${\mathsf{J}}_{{\mathsf{\varpi }}_2-}^{\mathsf{\eta }}\mathsf{\vartheta }\left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{({\mathsf{\varpi }}_2-\mathtt{x})}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{\zeta }\right)-{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\int }_{\mathtt{x}}^{{\mathsf{\varpi }}_2}{(\mathsf{\eta }-\mathtt{x})}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }.$$

□

• For any $x\in ({\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2)$, we have

$$\begin{array}{cc}& {\mathsf{J}}_{x-}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)+{\mathsf{J}}_{x+}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{\zeta }\right)\hfill \\ & ={\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}[{(x-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}+{({\mathsf{\varpi }}_2-\mathtt{x})}^{\mathsf{\eta }}]\mathsf{\vartheta }\left(x\right)\hfill \\ & +{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left[{\int }_{\mathtt{x}}^{{\mathsf{\varpi }}_2}{({\mathsf{\varpi }}_2-\mathsf{\eta })}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }-{\int }_{{\mathsf{\varpi }}_1}^x{(\mathsf{\eta }-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }\right].\hfill \end{array}$$

Proof. 

Since we have

$${\mathsf{J}}_{x+}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{\zeta }\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\eta }\right)}}{\int }_{\mathtt{x}}^{{\mathsf{\varpi }}_2}{({\mathsf{\varpi }}_2-\mathsf{\eta })}^{\mathsf{\eta }-1}\mathsf{\vartheta }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }$$

for ${\mathsf{\varpi }}_1⩽\mathtt{x}<{\mathsf{\varpi }}_2$ and

$${\mathsf{J}}_{x-}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\eta }\right)}}{\int }_{{\mathsf{\varpi }}_1}^x{(\mathsf{\eta }-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }-1}\mathsf{\vartheta }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }$$

for ${\mathsf{\varpi }}_1<x⩽{\mathsf{\varpi }}_2$. Since $\mathsf{\vartheta }:[{\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2]\to \mathsf{R}$ is an absolutely continuous function $[{\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2]$, then the integrals

$${\int }_{{\mathsf{\varpi }}_1}^x{(\mathsf{\eta }-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }\mathrm{and}{\int }_{\mathtt{x}}^{{\mathsf{\varpi }}_2}{({\mathsf{\varpi }}_2-\mathsf{\eta })}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }$$

exist, and integrating by parts, we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\int }_{{\mathsf{\varpi }}_1}^x{(\mathsf{\eta }-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }\hfill \\ \hfill & ={\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{(x-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}\mathsf{\vartheta }\left(x\right)-{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\eta }\right)}}{\int }_{{\mathsf{\varpi }}_1}^x{(\mathsf{\eta }-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }-1}\mathsf{\vartheta }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }\hfill \\ \hfill & ={\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{(x-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}\mathsf{\vartheta }\left(x\right)-{\mathsf{J}}_{x-}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)\hfill \end{array}$$

(6)

for ${\mathsf{\varpi }}_1<x⩽{\mathsf{\varpi }}_2$ and

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\int }_{\mathtt{x}}^{{\mathsf{\varpi }}_2}{({\mathsf{\varpi }}_2-\mathsf{\eta })}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }\hfill \\ \hfill & ={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\eta }\right)}}{\int }_{\mathtt{x}}^{{\mathsf{\varpi }}_2}{({\mathsf{\varpi }}_2-\mathsf{\eta })}^{\mathsf{\eta }-1}\mathsf{\vartheta }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }-{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{({\mathsf{\varpi }}_2-\mathtt{x})}^{\mathsf{\eta }}\mathsf{\vartheta }\left(x\right)\hfill \\ \hfill & ={\mathsf{J}}_{x+}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{\zeta }\right)-{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{({\mathsf{\varpi }}_2-\mathtt{x})}^{\mathsf{\eta }}\mathsf{\vartheta }\left(x\right)\hfill \end{array}$$

(7)

for ${\mathsf{\varpi }}_1⩽\mathtt{x}<{\mathsf{\varpi }}_2$. From (6), we have

$${\mathsf{J}}_{x-}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{(x-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}\mathsf{\vartheta }\left(x\right)-{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\int }_{{\mathsf{\varpi }}_1}^x{(\mathsf{\eta }-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }$$

for ${\mathsf{\varpi }}_1<x⩽{\mathsf{\varpi }}_2$ and from (7),

$${\mathsf{J}}_{x+}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{\zeta }\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{({\mathsf{\varpi }}_2-\mathtt{x})}^{\mathsf{\eta }}\mathsf{\vartheta }\left(x\right)+{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\int }_{\mathtt{x}}^{{\mathsf{\varpi }}_2}{({\mathsf{\varpi }}_2-\mathsf{\eta })}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }.$$

□

Corollary 1. 

Let $\mathsf{\vartheta }:[{\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2]\to \mathsf{R}$ be an absolutely continuous function on $[{\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2]$. We now possess the midpoint equality

$$\begin{array}{cc}& {\mathsf{J}}_{{\mathsf{\varpi }}_1+}^{\mathsf{\eta }}\mathsf{\vartheta }\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right)+{\mathsf{J}}_{{\mathsf{\varpi }}_2-}^{\mathsf{\eta }}\mathsf{\vartheta }\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right)\hfill \\ & ={\displaystyle \frac{1}{2^{\mathsf{\eta }-1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\displaystyle \frac{\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)+\mathsf{\vartheta }\left(\mathsf{\zeta }\right)}{2}}\hfill \\ & +{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left[{\int }_{{\mathsf{\varpi }}_1}^{{\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}}{\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}-\mathsf{\eta }\right)}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)d-{\int }_{{\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}}^{{\mathsf{\varpi }}_2}{\left(\mathsf{\eta }-{\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right)}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }\right]\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\mathsf{J}}_{{\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}-}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)+{\mathsf{J}}_{{\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}+}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{\zeta }\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2^{\mathsf{\eta }-1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\mathsf{\vartheta }\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right){({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left[{\int }_{{\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}}^{{\mathsf{\varpi }}_2}{({\mathsf{\varpi }}_2-\mathsf{\eta })}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }-{\int }_{{\mathsf{\varpi }}_1}^{{\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}}{(\mathsf{\eta }-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }\right],\hfill \end{array}$$

(8)

for ${\mathsf{\varpi }}_1⩽{\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}<{\mathsf{\varpi }}_2$. From (8), we have

$$\begin{array}{cc}\hfill & {\mathsf{J}}_{{\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}-}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)={\displaystyle \frac{1}{2^{\mathsf{\eta }-1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\mathsf{\vartheta }\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right){({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}-{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left[{\int }_{{\mathsf{\varpi }}_1}^{{\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}}{(\mathsf{\eta }-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }\right]\hfill \\ & ={\displaystyle \frac{1}{2^{\mathsf{\eta }-1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\mathsf{\vartheta }\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right){({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}-{\displaystyle \frac{{\mathsf{\delta }}^{\mathsf{\eta }}{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left[{\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right)\mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right],\hfill \end{array}$$

(9)

for ${\mathsf{\varpi }}_1<{\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}⩽{\mathsf{\varpi }}_2$, and from (8), we have

$$\begin{array}{cc}\hfill & {\mathsf{J}}_{{\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}+}^{\mathsf{\eta }}\mathsf{\vartheta }\left(\mathsf{\zeta }\right)={\displaystyle \frac{1}{2^{\mathsf{\eta }-1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\mathsf{\vartheta }\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right){({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}+{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left[{\int }_{{\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}}^{{\mathsf{\varpi }}_2}{({\mathsf{\varpi }}_2-\mathsf{\eta })}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }\right]\hfill \\ & ={\displaystyle \frac{1}{2^{\mathsf{\eta }-1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\mathsf{\vartheta }\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right){({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}-{\displaystyle \frac{{(1-\mathsf{\delta })}^{\mathsf{\eta }}{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left[{\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right)+{\mathsf{\varpi }}_2\mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right].\hfill \end{array}$$

(10)

4. Fractional Newton–Milne-Type Inequalities for Differentiable Convex Mappings

In this section, we employ many novel fractional identities to derive upper bounds for Newton–Milne-type inequalities involving differentiable convex mappings with different types of generalized convex mappings.

Lemma 4. 

Let the self-adjoint operators $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ have $\mathtt{Sp}\left(\mathsf{{\rm Y}}\right)\subset \mathsf{\Omega }$ and $\mathtt{Sp}\left(\mathsf{\zeta }\right)\subset$ Ω. Assume that $\mathsf{\vartheta }$ is a continuous and convex function on $\mathsf{\Omega },$ and then one has

$$\begin{array}{cc}\hfill & \left[{\displaystyle \frac{2}{3}}(\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)\otimes 1)-{\displaystyle \frac{1}{3}}\mathsf{\vartheta }\left({\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}\right)+{\displaystyle \frac{2}{3}}(1\otimes \mathsf{\vartheta }\left(\mathsf{\zeta }\right))\right]-{\displaystyle \frac{\mathsf{\Gamma }(\mathsf{\eta }+1)}{{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}}}\left[{\displaystyle \frac{1}{2^{\mathsf{\eta }-1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\displaystyle \frac{\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)+\mathsf{\vartheta }\left(\mathsf{\zeta }\right)}{2}}\right.\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left[{\displaystyle \frac{{(1-\mathsf{\delta })}^{\mathsf{\eta }}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left[{\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+\mathsf{\zeta }\otimes 1}{2}}\right)+1\otimes \mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right]\right.\hfill \\ \hfill & \left.\left.+{\displaystyle \frac{{\mathsf{\delta }}^{\mathsf{\eta }}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left[{\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+1\otimes \mathsf{\zeta }}{2}}\right)\mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right]\right]\right]\hfill \\ \hfill =& {\displaystyle \frac{1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1}{2}}\left[{\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }\mathsf{\zeta }\otimes 1\right)+{\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }\mathsf{\zeta }\otimes 1\right)\right.\hfill \\ \hfill & \left.+{\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }\mathsf{\zeta }\otimes 1\right)-{\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }\mathsf{\zeta }\otimes 1\right)\right]\mathtt{d}\mathsf{\delta }\hfill \end{array}$$

(11)

Proof. 

Considering the following finding [81] for a fractional integral operator based on differentiable convex mappings:

Let $\mathsf{\vartheta }:[{\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2]\to \mathsf{R}$ be a differentiable mapping $({\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2)$ such that ${\mathsf{\vartheta }}^{\prime }\in {\mathtt{L}}_{\mathtt{1}}\left([{\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2]\right)$. Then, the following equality holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{3}}\left[2\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)-\mathsf{\vartheta }\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right)+2\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)\right]-{\displaystyle \frac{\mathsf{\Gamma }(\mathsf{\eta }+1)}{{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}}}\left[{\mathsf{J}}_{{\mathsf{\varpi }}_2-}^{\mathsf{\eta }}\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)+{\mathsf{J}}_{{\mathsf{\varpi }}_1+}^{\mathsf{\eta }}\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)\right]\hfill \\ \hfill =& {\displaystyle \frac{{\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1}{2}}\left[{\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2\right)+{\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2\right)\right.\hfill \\ \hfill & \left.+{\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2\right)-{\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2\right)\right]\mathtt{d}\mathsf{\delta }\hfill \end{array}$$

(12)

Taking into account Lemma 8 and Corollary 1 and applying them to Equation (12), one has

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{3}}\left[2\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)-\mathsf{\vartheta }\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right)+2\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)\right]-{\displaystyle \frac{\mathsf{\Gamma }(\mathsf{\eta }+1)}{{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}}}\left[{\mathsf{J}}_{{\mathsf{\varpi }}_2-}^{\mathsf{\eta }}\mathsf{\vartheta }\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right)+{\mathsf{J}}_{{\mathsf{\varpi }}_1+}^{\mathsf{\eta }}\mathsf{\vartheta }\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right)\right]\hfill \\ \hfill =& \left[{\displaystyle \frac{2}{3}}\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)-{\displaystyle \frac{1}{3}}\mathsf{\vartheta }\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right)+{\displaystyle \frac{2}{3}}\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)\right]-{\displaystyle \frac{\mathsf{\Gamma }(\mathsf{\eta }+1)}{{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}}}\left[{\displaystyle \frac{1}{2^{\mathsf{\eta }-1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\displaystyle \frac{\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)+\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)}{2}}\right.\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left[{\int }_{{\mathsf{\varpi }}_1}^{{\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}}{\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}-\mathsf{\eta }\right)}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }-{\int }_{{\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}}^{{\mathsf{\varpi }}_2}{\left(\mathsf{\eta }-{\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right)}^{\mathsf{\eta }}{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\mathtt{d}\mathsf{\eta }\right.]\hfill \\ \hfill =& \left[{\displaystyle \frac{2}{3}}\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)-{\displaystyle \frac{1}{3}}\mathsf{\vartheta }\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right)+{\displaystyle \frac{2}{3}}\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)\right]-{\displaystyle \frac{\mathsf{\Gamma }(\mathsf{\eta }+1)}{{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}}}\left[{\displaystyle \frac{1}{2^{\mathsf{\eta }-1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\displaystyle \frac{\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)+\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)}{2}}\right.\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left[{\displaystyle \frac{{(1-\mathsf{\delta })}^{\mathsf{\eta }}{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left[{\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right)+{\mathsf{\varpi }}_2\mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right]\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{{\mathsf{\delta }}^{\mathsf{\eta }}{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left[{\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right)\mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right]\right]\hfill \\ \hfill =& {\displaystyle \frac{{\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1}{2}}\left[{\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2\right)+{\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2\right)\right.\hfill \\ \hfill & \left.+{\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2\right)-{\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2\right)\right]\mathtt{d}\mathsf{\delta }\hfill \end{array}$$

(13)

Assume that $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ have the spectral resolutions

$$\mathsf{{\rm Y}}={\int }_{\mathsf{\Omega }}{\mathsf{\varpi }}_2\mathtt{dE}\left({\mathsf{\varpi }}_2\right)\mathrm{and}\mathsf{\zeta }={\int }_{\mathsf{\Omega }}{\mathsf{\varpi }}_1\mathtt{dF}\left({\mathsf{\varpi }}_1\right).$$

If we take the integral ${\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}$ over ${\mathtt{dE}}_{{\mathsf{\varpi }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\varpi }}_2}$ in (13), then we obtain

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left[{\displaystyle \frac{2}{3}}\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)-{\displaystyle \frac{1}{3}}\mathsf{\vartheta }({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}})+{\displaystyle \frac{2}{3}}\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)\right]\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}-{\displaystyle \frac{\mathsf{\Gamma }(\mathsf{\eta }+1)}{{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}}}{\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left[{\displaystyle \frac{1}{2^{\mathsf{\eta }-1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\displaystyle \frac{\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)+\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)}{2}}\right.\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\displaystyle \frac{{(1-\mathsf{\delta })}^{\mathsf{\eta }}{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\int }_0^1{\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta })({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}})+{\mathsf{\varpi }}_2\mathsf{\delta })\mathtt{d}\mathsf{\delta }\hfill \\ \hfill & +{\displaystyle \frac{{\mathsf{\delta }}^{\mathsf{\eta }}{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\int }_0^1{\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta }){\mathsf{\varpi }}_1+({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}})\mathsf{\delta })\mathtt{d}\mathsf{\delta }]\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}\hfill \\ \hfill =& {\displaystyle \frac{{\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1}{2}}{\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}[{\int }_0^{{\displaystyle \frac{1}{2}}}({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}){\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2)+{\int }_{{\displaystyle \frac{1}{2}}}^1({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}){\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2)\hfill \\ \hfill & +{\int }_0^{{\displaystyle \frac{1}{2}}}({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}){\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2)-{\int }_{{\displaystyle \frac{1}{2}}}^1({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}){\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2)]\mathtt{d}\mathsf{\delta }\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}.\hfill \end{array}$$

(14)

Taking into account Lemma 1 and Fubini’s theorem, we obtain the following representations:

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}=(\mathsf{\vartheta }\left(\mathsf{\zeta }\right)\otimes 1),\hfill \\ \hfill & {\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\mathsf{\vartheta }\left({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}}\right)\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}=\mathsf{\vartheta }\left({\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}\right),\hfill \\ \hfill & {\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}=(1\otimes \mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)).\hfill \end{array}$$

(15)

A similar approach can be taken to the right side as well that is

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1}{2}}{\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left[{\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2\right)+{\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2\right)\right.\hfill \\ \hfill & \left.+{\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2\right)-{\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2\right)\right]\mathtt{d}\mathsf{\delta }\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}\hfill \\ & ={\displaystyle \frac{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}{2}}\left[{\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)+{\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)\right.\hfill \\ \hfill & \left.+{\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)-{\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)\right]\mathtt{d}\mathsf{\delta }\hfill \end{array}$$

(16)

Using Equations (15) and (16) in (14), we obtain the required result. □

Now we use Lemma 4 with different forms of generalized convex mappings to establish the upper bounds.

Theorem 7. 

Let the self-adjoint operators $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ have $\mathtt{Sp}\left(\mathsf{{\rm Y}}\right)\subset \mathsf{\Omega }$ and $\mathtt{Sp}\left(\mathsf{\zeta }\right)\subset$ Ω. Assume that $\mathsf{\vartheta }$ is differentiable on Ω with ${∥{\mathsf{\vartheta }}^{\prime }∥}_{\mathsf{\Omega },\infty }:=$ ${sup}_{\mathsf{\eta }\in \mathsf{\Omega }}\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\eta }\right)\right|<\infty$; then, one has

$$\begin{array}{cc}\hfill & \parallel \left({\displaystyle \frac{2}{3}}(\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)\otimes 1)-{\displaystyle \frac{1}{3}}\mathsf{\vartheta }\left({\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}\right)+{\displaystyle \frac{2}{3}}(1\otimes \mathsf{\vartheta }\left(\mathsf{\zeta }\right))\right)-{\displaystyle \frac{\mathsf{\Gamma }(\mathsf{\eta }+1)}{{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}}}({\displaystyle \frac{1}{2^{\mathsf{\eta }-1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\displaystyle \frac{\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)+\mathsf{\vartheta }\left(\mathsf{\zeta }\right)}{2}}\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\displaystyle \frac{{(1-\mathsf{\delta })}^{\mathsf{\eta }}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\int }_0^1{\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta })\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+\mathsf{\zeta }\otimes 1}{2}}\right)+1\otimes \mathsf{\delta })\mathtt{d}\mathsf{\delta })\hfill \\ \hfill & +{\displaystyle \frac{{\mathsf{\delta }}^{\mathsf{\eta }}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\int }_0^1{\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+1\otimes \mathsf{\zeta }}{2}}\right)\mathsf{\delta })\mathtt{d}\mathsf{\delta }))\parallel \hfill \\ & \le {\displaystyle \frac{\parallel 1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1\parallel }{2}}\left(2\parallel {\mathsf{\vartheta }}^{\prime }{\parallel }_{\mathsf{\Omega },+\infty }\left({\displaystyle \frac{(2\mathsf{\eta }+2)·2^{\mathsf{\eta }}-3}{(6\mathsf{\eta }+6)·2^{\mathsf{\eta }}}}\right)+2{\parallel {\mathsf{\vartheta }}^{\prime }\parallel }_{\mathsf{\Omega },+\infty }\left({\displaystyle \frac{(\mathsf{\eta }+5)·2^{\mathsf{\eta }}+3}{(6\mathsf{\eta }+6)·2^{\mathsf{\eta }}}}\right)\right).\hfill \end{array}$$

Proof. 

Using the triangle inequality and the operator norm from Lemma 4, we obtain

$$\begin{array}{cc}\hfill & \parallel \left({\displaystyle \frac{2}{3}}(\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)\otimes 1)-{\displaystyle \frac{1}{3}}\mathsf{\vartheta }\left({\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}\right)+{\displaystyle \frac{2}{3}}(1\otimes \mathsf{\vartheta }\left(\mathsf{\zeta }\right))\right)-{\displaystyle \frac{\mathsf{\Gamma }(\mathsf{\eta }+1)}{{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}}}\left({\displaystyle \frac{1}{2^{\mathsf{\eta }-1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\displaystyle \frac{\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)+\mathsf{\vartheta }\left(\mathsf{\zeta }\right)}{2}}\right.\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left({\displaystyle \frac{{(1-\mathsf{\delta })}^{\mathsf{\eta }}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left({\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+\mathsf{\zeta }\otimes 1}{2}}\right)+1\otimes \mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right)\right.\hfill \\ \hfill & +{\displaystyle \frac{{\mathsf{\delta }}^{\mathsf{\eta }}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left({\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+1\otimes \mathsf{\zeta }}{2}}\right)\mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right))\parallel \hfill \\ & \le {\displaystyle \frac{\parallel 1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1\parallel }{2}}\parallel [{\int }_0^{{\displaystyle \frac{1}{2}}}({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}){\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1)+{\int }_{{\displaystyle \frac{1}{2}}}^1({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}){\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1)\hfill \\ \hfill & +{\int }_0^{{\displaystyle \frac{1}{2}}}({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}){\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1)-{\int }_{{\displaystyle \frac{1}{2}}}^1({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}){\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1)]\mathtt{d}\mathsf{\delta }\parallel \hfill \\ & \le {\displaystyle \frac{\parallel 1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1\parallel }{2}}(\parallel {\int }_0^{{\displaystyle \frac{1}{2}}}({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}){\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1)\mathtt{d}\mathsf{\delta }\parallel +\parallel {\int }_{{\displaystyle \frac{1}{2}}}^1({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}){\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1)\mathtt{d}\mathsf{\delta }\parallel \hfill \\ \hfill & +\parallel {\int }_0^{{\displaystyle \frac{1}{2}}}({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}){\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1)\mathtt{d}\mathsf{\delta }\parallel +\parallel {\int }_{{\displaystyle \frac{1}{2}}}^1({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}){\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1)\mathtt{d}\mathsf{\delta }\parallel ).\hfill \end{array}$$

(17)

Considering Lemma 1, we obtain

$$\begin{array}{cc}\hfill & \left|\left({\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1+\left(1-\mathsf{\delta }\right)1\otimes \mathsf{\zeta }\right)\right|={\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left|\left({\mathsf{\vartheta }}^{\prime }\mathsf{\delta }{\mathsf{\varpi }}_1+\left(1-\mathsf{\delta }\right){\mathsf{\varpi }}_2\right)\right|\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}.\right.\hfill \end{array}$$

Since

$$\left|\left({\mathsf{\vartheta }}^{\prime }(\mathsf{\delta }{\mathsf{\varpi }}_1+\left(1-\mathsf{\delta }\right){\mathsf{\varpi }}_2\right)\right|⩽{∥{\mathsf{\vartheta }}^{\prime }∥}_{\mathsf{\Omega },+\infty }$$

for all $\mathsf{\delta }\in [0,1]$ and ${\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2\in \mathsf{\Omega }$.

If we take the integral ${\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}$ over ${\mathtt{dE}}_{{\mathsf{\varpi }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\varpi }}_2}$, then we obtain

$$\begin{array}{cc}\hfill & \left|\left({\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1+\left(1-\mathsf{\delta }\right)1\otimes \mathsf{\zeta }\right)\right|={\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left|\left({\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }{\mathsf{\varpi }}_1\right)+\left(1-\mathsf{\delta }\right){\mathsf{\varpi }}_2\right)\right|\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}\right.\hfill \\ \hfill & ⩽{∥{\mathsf{\vartheta }}^{\prime }∥}_{\mathsf{\Omega },+\infty }{\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}={∥{\mathsf{\vartheta }}^{\prime }∥}_{\mathsf{\Omega },+\infty }.\hfill \end{array}$$

(18)

Similarly, we obtain

$$\begin{array}{cc}\hfill & \left|\right({\mathsf{\vartheta }}^{\prime }(\mathsf{\delta }({\displaystyle \frac{1\otimes \mathsf{\zeta }+\mathsf{{\rm Y}}\otimes 1}{2}})+\left(1-\mathsf{\delta }\right)\mathsf{{\rm Y}}\otimes 1)|={\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left|\right({\mathsf{\vartheta }}^{\prime }(\mathsf{\delta }({\displaystyle \frac{{\mathsf{\varpi }}_1+{\mathsf{\varpi }}_2}{2}})+(1-\mathsf{\delta }){\mathsf{\varpi }}_1)|\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}\hfill \\ \hfill & ⩽\parallel {\mathsf{\vartheta }}^{\prime }{\parallel }_{\mathsf{\Omega },+\infty }{\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}={\parallel {\mathsf{\vartheta }}^{\prime }\parallel }_{\mathsf{\Omega },+\infty }.\hfill \end{array}$$

(19)

Also, we have

$$\begin{array}{c}\hfill \parallel {\int }_0^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right)\mathtt{d}\mathsf{\delta }\parallel =\left({\displaystyle \frac{\left(2\mathsf{\eta }+2\right)·2^{\mathsf{\eta }}-3}{\left(6\mathsf{\eta }+6\right)·2^{\mathsf{\eta }}}}\right),\\ \hfill \parallel {\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right)\mathtt{d}\mathsf{\delta }\parallel =\left({\displaystyle \frac{\left(\mathsf{\eta }+5\right)·2^{\mathsf{\eta }}+3}{\left(6\mathsf{\eta }+6\right)·2^{\mathsf{\eta }}}}\right).\end{array}$$

(20)

Using Equations (18)–(20) in (17), we obtain the required result. □

Theorem 8. 

Let the self-adjoint operators $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ have $\mathtt{Sp}\left(\mathsf{{\rm Y}}\right)\subset \mathsf{\Omega }$ and $\mathtt{Sp}\left(\mathsf{\zeta }\right)\subset$ Ω. Assume that $\mathsf{\vartheta }$ is differentiable on Ω with $\left|{\mathsf{\vartheta }}^{\prime }\right|$ convex on Ω; then, one has

$$\begin{array}{cc}\hfill & \parallel \left({\displaystyle \frac{2}{3}}(\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)\otimes 1)-{\displaystyle \frac{1}{3}}\mathsf{\vartheta }\left({\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}\right)+{\displaystyle \frac{2}{3}}(1\otimes \mathsf{\vartheta }\left(\mathsf{\zeta }\right))\right)-{\displaystyle \frac{\mathsf{\Gamma }(\mathsf{\eta }+1)}{{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}}}\left({\displaystyle \frac{1}{2^{\mathsf{\eta }-1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\displaystyle \frac{\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)+\mathsf{\vartheta }\left(\mathsf{\zeta }\right)}{2}}\right.\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left({\displaystyle \frac{{(1-\mathsf{\delta })}^{\mathsf{\eta }}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left({\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+\mathsf{\zeta }\otimes 1}{2}}\right)+1\otimes \mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right)\right.\hfill \\ \hfill & +{\displaystyle \frac{{\mathsf{\delta }}^{\mathsf{\eta }}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left({\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+1\otimes \mathsf{\zeta }}{2}}\right)\mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right))\parallel \hfill \\ \hfill & \le {\displaystyle \frac{\parallel 1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1\parallel }{2}}\left(\left({\displaystyle \frac{7{\mathsf{\eta }}^2+21\mathsf{\eta }-3·2^{1-\mathsf{\eta }}\mathsf{\eta }-9·2^{1-\mathsf{\eta }}+72·2^{-\mathsf{\eta }-2}+24\mathsf{\eta }·2^{-\mathsf{\eta }-2}-10}{12(\mathsf{\eta }+1)(\mathsf{\eta }+2)}}\right)∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\otimes 1∥\right.\hfill \\ \hfill & \left.+\left({\displaystyle \frac{24·2^{-\mathsf{\eta }-2}-3·2^{1-\mathsf{\eta }}+5\mathsf{\eta }-14}{12(\mathsf{\eta }+2)}}\right)∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\otimes 1∥\right).\hfill \end{array}$$

Proof. 

As $\left|{\mathsf{\vartheta }}^{\prime }\right|$ is convex on $\mathsf{\Omega }$, we obtain

$$\begin{array}{cc}\hfill & \left.\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }{\mathsf{\varpi }}_1\right)+\left(1-\mathsf{\delta }\right){\mathsf{\varpi }}_2\right)\right|\le \mathsf{\delta }\left|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_2\right)\right|+\left(1-\mathsf{\delta }\right)\left|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_1\right)\right|\hfill \end{array}$$

for all $\mathsf{\delta }\in [0,1]$ and ${\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2\in \mathsf{\Omega }$.

If we take the integral ${\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}$ over ${\mathtt{dE}}_{{\mathsf{\varpi }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\varpi }}_2}$, then we obtain

$$\begin{array}{cc}\hfill & \left|\left({\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }\left(\mathsf{{\rm Y}}\otimes 1\right)+\left(1-\mathsf{\delta }\right)1\otimes \mathsf{\zeta }\right)\right|={\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left|\left({\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }\left({\mathsf{\varpi }}_1\right)+\left(1-\mathsf{\delta }\right){\mathsf{\varpi }}_2\right)\right|\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}\right.\right.\hfill \\ \hfill & \le (1-\mathsf{\delta }){\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_1\right)\right|\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}+\mathsf{\delta }{\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_2\right)\right|\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2},\hfill \end{array}$$

(21)

namely

$$\begin{array}{c}\hfill \left|{\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }1\otimes \mathsf{\zeta })\right|\le (1-\mathsf{\delta })\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+\mathsf{\delta }\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|\otimes 1\end{array}$$

(22)

for all $\mathsf{\delta }\in [0,1]$.

If we take the norm in (22), then we obtain

$$\begin{array}{cc}\hfill ∥{\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }1\otimes \mathsf{\zeta })∥& \le ∥(1-\mathsf{\delta })\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+\mathsf{\delta }\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|\otimes 1∥\hfill \\ & \le (1-\mathsf{\delta })∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)∥+\mathsf{\delta }∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)∥\hfill \end{array}$$

(23)

Applying the triangle inequality to the operator norm in (11), we obtain

$$\begin{array}{cc}\hfill & \parallel \left({\displaystyle \frac{2}{3}}(\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)\otimes 1)-{\displaystyle \frac{1}{3}}\mathsf{\vartheta }\left({\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}\right)+{\displaystyle \frac{2}{3}}(1\otimes \mathsf{\vartheta }\left(\mathsf{\zeta }\right))\right)-{\displaystyle \frac{\mathsf{\Gamma }(\mathsf{\eta }+1)}{{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}}}\left({\displaystyle \frac{1}{2^{\mathsf{\eta }-1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\displaystyle \frac{\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)+\mathsf{\vartheta }\left(\mathsf{\zeta }\right)}{2}}\right.\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left({\displaystyle \frac{{(1-\mathsf{\delta })}^{\mathsf{\eta }}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left({\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+\mathsf{\zeta }\otimes 1}{2}}\right)+1\otimes \mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right)\right.\hfill \\ \hfill & +{\displaystyle \frac{{\mathsf{\delta }}^{\mathsf{\eta }}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left({\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+1\otimes \mathsf{\zeta }}{2}}\right)\mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right))\parallel \hfill \\ & \le {\displaystyle \frac{\parallel 1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1\parallel }{2}}\parallel \left[{\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)+{\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)\right.\hfill \\ \hfill & \left.+{\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)-{\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)\right]\mathtt{d}\mathsf{\delta }\parallel \hfill \\ & \le {\displaystyle \frac{\parallel 1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1\parallel }{2}}\left(\parallel {\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)\mathtt{d}\mathsf{\delta }\parallel +\parallel {\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)\mathtt{d}\mathsf{\delta }\parallel \right.\hfill \\ \hfill & +\parallel {\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)\mathtt{d}\mathsf{\delta }\parallel +\parallel {\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)\mathtt{d}\mathsf{\delta }\parallel )\hfill \\ & \le {\displaystyle \frac{\parallel 1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1\parallel }{2}}\left(\parallel {\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right)\left((1-\mathsf{\delta })\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+\mathsf{\delta }\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|\otimes 1\mathtt{d}\mathsf{\delta }\right)\parallel \right.\hfill \\ & +\parallel {\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right)\left((1-\mathsf{\delta })\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+\mathsf{\delta }\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|\otimes 1\mathtt{d}\mathsf{\delta }\right)\parallel \hfill \\ \hfill & +\parallel {\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right)\left((1-\mathsf{\delta })\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+\mathsf{\delta }\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|\otimes 1\mathtt{d}\mathsf{\delta }\right)\parallel \hfill \\ & \left.+\parallel {\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right)\left((1-\mathsf{\delta })\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+\mathsf{\delta }\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|\otimes 1\right)\mathtt{d}\mathsf{\delta }\right)\parallel )\hfill \\ & \le {\displaystyle \frac{\parallel 1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1\parallel }{2}}\left(\parallel \left({\displaystyle \frac{\left({\mathsf{\eta }}^2+3\mathsf{\eta }+2\right)·2^{\mathsf{\eta }}-\mathsf{\eta }-3}{\left(4{\mathsf{\eta }}^2+12\mathsf{\eta }+8\right)·2^{\mathsf{\eta }}}}\right)\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+\left({\displaystyle \frac{\left(\mathsf{\eta }+2\right)·2^{\mathsf{\eta }}-3}{\left(12\mathsf{\eta }+24\right)·2^{\mathsf{\eta }}}}\right)\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|\otimes 1\parallel \right.\hfill \\ & +\parallel \left({\displaystyle \frac{\left({\mathsf{\eta }}^2+3\mathsf{\eta }-22\right)·2^{\mathsf{\eta }}+6\mathsf{\eta }+18}{\left(24{\mathsf{\eta }}^2+72\mathsf{\eta }+48\right)·2^{\mathsf{\eta }}}}\right)\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+\left({\displaystyle \frac{\left(\mathsf{\eta }-6\right)·2^{\mathsf{\eta }}+2}{\left(8\mathsf{\eta }+16\right)·2^{\mathsf{\eta }}}}\right)\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|\otimes 1\parallel \hfill \\ \hfill & +\parallel \left({\displaystyle \frac{\left({\mathsf{\eta }}^2+3\mathsf{\eta }+2\right)·2^{\mathsf{\eta }}-\mathsf{\eta }-3}{\left(4{\mathsf{\eta }}^2+12\mathsf{\eta }+8\right)·2^{\mathsf{\eta }}}}\right)\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+\left({\displaystyle \frac{\left(\mathsf{\eta }+2\right)·2^{\mathsf{\eta }}-3}{\left(12\mathsf{\eta }+24\right)·2^{\mathsf{\eta }}}}\right)\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|\otimes 1\parallel \hfill \\ & \left.+\parallel \left({\displaystyle \frac{\left({\mathsf{\eta }}^2+3\mathsf{\eta }-22\right)·2^{\mathsf{\eta }}+6\mathsf{\eta }+18}{\left(24{\mathsf{\eta }}^2+72\mathsf{\eta }+48\right)·2^{\mathsf{\eta }}}}\right)\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+\left({\displaystyle \frac{\left(\mathsf{\eta }-6\right)·2^{\mathsf{\eta }}+2}{\left(8\mathsf{\eta }+16\right)·2^{\mathsf{\eta }}}}\right)\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|\otimes 1\parallel \right)\hfill \\ & \le {\displaystyle \frac{\parallel 1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1\parallel }{2}}\left(\left({\displaystyle \frac{7{\mathsf{\eta }}^2+21\mathsf{\eta }-3·2^{1-\mathsf{\eta }}\mathsf{\eta }-9·2^{1-\mathsf{\eta }}+72·2^{-\mathsf{\eta }-2}+24\mathsf{\eta }·2^{-\mathsf{\eta }-2}-10}{12(\mathsf{\eta }+1)(\mathsf{\eta }+2)}}\right)∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\otimes 1∥\right.\hfill \\ & \left.+\left({\displaystyle \frac{24·2^{-\mathsf{\eta }-2}-3·2^{1-\mathsf{\eta }}+5\mathsf{\eta }-14}{12(\mathsf{\eta }+2)}}\right)∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\otimes 1∥\right).\hfill \end{array}$$

(24)

Recall that $\mathsf{\Phi }:\mathsf{\Omega }\to \mathsf{R}$ is a quasi-convex function if

$$\mathsf{\Phi }((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2)\le max\{\mathsf{\Phi }\left({\mathsf{\varpi }}_2\right),\mathsf{\Phi }\left({\mathsf{\varpi }}_1\right)\}={\displaystyle \frac{1}{2}}(\mathsf{\Phi }\left({\mathsf{\varpi }}_2\right)+\mathsf{\Phi }\left({\mathsf{\varpi }}_1\right)+|\mathsf{\Phi }\left({\mathsf{\varpi }}_2\right)-\mathsf{\Phi }\left({\mathsf{\varpi }}_1\right)\left|\right)$$

for all ${\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2\in \mathsf{\Omega }$, and $\mathsf{\delta }\in [0,1]$. □

Theorem 9. 

Let the self-adjoint operators $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ have $\mathtt{Sp}\left(\mathsf{{\rm Y}}\right)\subset \mathsf{\Omega }$ and $\mathtt{Sp}\left(\mathsf{\zeta }\right)\subset$ Ω. Assume that $\mathsf{\vartheta }$ is continuously differentiable on Ω with $\left|{\mathsf{\vartheta }}^{\prime }\right|$ being quasi-convex on Ω; then, one has

$$\begin{array}{cc}\hfill & \parallel \left({\displaystyle \frac{2}{3}}(\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)\otimes 1)-{\displaystyle \frac{1}{3}}\mathsf{\vartheta }\left({\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}\right)+{\displaystyle \frac{2}{3}}(1\otimes \mathsf{\vartheta }\left(\mathsf{\zeta }\right))\right)-{\displaystyle \frac{\mathsf{\Gamma }(\mathsf{\eta }+1)}{{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}}}\left({\displaystyle \frac{1}{2^{\mathsf{\eta }-1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\displaystyle \frac{\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)+\mathsf{\vartheta }\left(\mathsf{\zeta }\right)}{2}}\right.\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left({\displaystyle \frac{{(1-\mathsf{\delta })}^{\mathsf{\eta }}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left({\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+\mathsf{\zeta }\otimes 1}{2}}\right)+1\otimes \mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right)\right.\hfill \\ \hfill & +{\displaystyle \frac{{\mathsf{\delta }}^{\mathsf{\eta }}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left({\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+1\otimes \mathsf{\zeta }}{2}}\right)\mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right))\parallel \hfill \\ \hfill & \le {\displaystyle \frac{\parallel 1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1\parallel }{2}}\left(\left({\displaystyle \frac{\left(2\mathsf{\eta }+2\right)·2^{\mathsf{\eta }}-3}{\left(6\mathsf{\eta }+6\right)·2^{\mathsf{\eta }}}}\right)+\left({\displaystyle \frac{\left(\mathsf{\eta }+5\right)·2^{\mathsf{\eta }}+3}{\left(6\mathsf{\eta }+6\right)·2^{\mathsf{\eta }}}}\right)\right)\parallel \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)|\otimes 1-1\otimes |{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\left|\right|\parallel ).\hfill \end{array}$$

Proof. 

As $\left|{\mathsf{\vartheta }}^{\prime }\right|$ is quasi-convex on $\mathsf{\Omega }$, one has

$$\begin{array}{cc}\hfill & \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }\left({\mathsf{\varpi }}_1\right)+\left(1-\mathsf{\delta }\right){\mathsf{\varpi }}_2\right)\right|\le {\displaystyle \frac{1}{2}}\left(\left|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_2\right)\right|+\left|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_1\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_2\right)|-|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_1\right)\left|\right|\right)\hfill \end{array}$$

for all $\mathsf{\delta }\in [0,1]$ and ${\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2\in \mathsf{\Omega }$.

Taking the double integral ${\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}$ over ${\mathtt{dE}}_{{\mathsf{\varpi }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\varpi }}_2}$ yields

$$\begin{array}{cc}\hfill & \left|\left({\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }\left(\mathsf{{\rm Y}}\otimes 1\right)+\left(1-\mathsf{\delta }\right)1\otimes \mathsf{\zeta }\right)\right|\right.\hfill \\ \hfill & ={\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left|\left({\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }\left({\mathsf{\varpi }}_2\right)+\left(1-\mathsf{\delta }\right){\mathsf{\varpi }}_2\right)\right|\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}\right.\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left(\left|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_2\right)\right|+\left|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_1\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_2\right)|-|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_1\right)\left|\right|\right)\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}\left(\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)|\otimes 1-1\otimes |{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\left|\right|\right).\hfill \end{array}$$

for all $\mathsf{\delta }\in [0,1]$.

Using the above inequality’s norm, the following results are obtained:

$$\begin{array}{cc}\hfill & ∥\left({\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }\left(\mathsf{{\rm Y}}\otimes 1\right)+\left(1-\mathsf{\delta }\right)1\otimes \mathsf{\zeta }\right)\right.∥\hfill \\ \hfill & \le ∥{\displaystyle \frac{1}{2}}\left(\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)|\otimes 1-1\otimes |{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\left|\right|\right)∥\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{2}}\left(∥\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|∥+∥\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1-1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|∥\right).\hfill \end{array}$$

Applying the triangle inequality to the operator norm in (11), we obtain

$$\begin{array}{cc}\hfill & \parallel \left({\displaystyle \frac{2}{3}}(\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)\otimes 1)-{\displaystyle \frac{1}{3}}\mathsf{\vartheta }\left({\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}\right)+{\displaystyle \frac{2}{3}}(1\otimes \mathsf{\vartheta }\left(\mathsf{\zeta }\right))\right)-{\displaystyle \frac{\mathsf{\Gamma }(\mathsf{\eta }+1)}{{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\mathsf{\eta }}}}\left({\displaystyle \frac{1}{2^{\mathsf{\eta }-1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}{\displaystyle \frac{\mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)+\mathsf{\vartheta }\left(\mathsf{\zeta }\right)}{2}}\right.\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left({\displaystyle \frac{{(1-\mathsf{\delta })}^{\mathsf{\eta }}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left({\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+\mathsf{\zeta }\otimes 1}{2}}\right)+1\otimes \mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right)\right.\hfill \\ \hfill & +{\displaystyle \frac{{\mathsf{\delta }}^{\mathsf{\eta }}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\mathsf{\eta }+1}}{2^{\mathsf{\eta }+1}\mathsf{\Gamma }(\mathsf{\eta }+1)}}\left({\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+1\otimes \mathsf{\zeta }}{2}}\right)\mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right))\parallel \hfill \\ \hfill & \le {\displaystyle \frac{\parallel 1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1\parallel }{2}}\parallel \left[{\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)+{\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)\right.\hfill \\ \hfill & \left.+{\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)-{\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)\right]\mathtt{d}\mathsf{\delta }\parallel \hfill \\ & \le {\displaystyle \frac{\parallel 1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1\parallel }{2}}\left(\parallel {\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)\mathtt{d}\mathsf{\delta }\parallel +\parallel {\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)\mathtt{d}\mathsf{\delta }\parallel \right.\hfill \\ \hfill & +\parallel {\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)\mathtt{d}\mathsf{\delta }\parallel +\parallel {\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1\right)\mathtt{d}\mathsf{\delta }\parallel )\hfill \\ \hfill & \le {\displaystyle \frac{\parallel 1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1\parallel }{2}}\left(\parallel {\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right)\left({\displaystyle \frac{1}{2}}\left(\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)|\otimes 1-1\otimes |{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\left|\right|\right)\right)\parallel \right.\hfill \\ & +\parallel {\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right)\left({\displaystyle \frac{1}{2}}\left(\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)|\otimes 1-1\otimes |{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\left|\right|\right)\right)\parallel \hfill \\ \hfill & +\parallel {\int }_0^{{\displaystyle \frac{1}{2}}}\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{2}{3}}\right)\left({\displaystyle \frac{1}{2}}\left(\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)|\otimes 1-1\otimes |{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\left|\right|\right)\right)\parallel \hfill \\ \hfill & +\parallel {\int }_{{\displaystyle \frac{1}{2}}}^1\left({\mathsf{\delta }}^{\mathsf{\eta }}-{\displaystyle \frac{1}{3}}\right)\left({\displaystyle \frac{1}{2}}\left(\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)|\otimes 1-1\otimes |{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\left|\right|\right)\right)\parallel )\hfill \\ \hfill & \le {\displaystyle \frac{\parallel 1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1\parallel }{2}}\left(\parallel \left({\displaystyle \frac{\left(2\mathsf{\eta }+2\right)·2^{\mathsf{\eta }}-3}{\left(6\mathsf{\eta }+6\right)·2^{\mathsf{\eta }}}}\right)\left({\displaystyle \frac{1}{2}}\left(\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)|\otimes 1-1\otimes |{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\left|\right|\right)\right)\parallel \right.\hfill \\ & +\parallel \left({\displaystyle \frac{\left(\mathsf{\eta }+5\right)·2^{\mathsf{\eta }}+3}{\left(6\mathsf{\eta }+6\right)·2^{\mathsf{\eta }}}}\right)\left({\displaystyle \frac{1}{2}}\left(\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)|\otimes 1-1\otimes |{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\left|\right|\right)\right)\parallel \hfill \\ \hfill & +\parallel \left({\displaystyle \frac{\left(2\mathsf{\eta }+2\right)·2^{\mathsf{\eta }}-3}{\left(6\mathsf{\eta }+6\right)·2^{\mathsf{\eta }}}}\right)\left({\displaystyle \frac{1}{2}}\left(\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)|\otimes 1-1\otimes |{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\left|\right|\right)\right)\parallel \hfill \\ \hfill & +\parallel \left({\displaystyle \frac{\left(\mathsf{\eta }+5\right)·2^{\mathsf{\eta }}+3}{\left(6\mathsf{\eta }+6\right)·2^{\mathsf{\eta }}}}\right)\left({\displaystyle \frac{1}{2}}\left(\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)|\otimes 1-1\otimes |{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\left|\right|\right)\right)\parallel )\hfill \\ \hfill & \le {\displaystyle \frac{\parallel 1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1\parallel }{2}}\left(\left({\displaystyle \frac{\left(2\mathsf{\eta }+2\right)·2^{\mathsf{\eta }}-3}{\left(6\mathsf{\eta }+6\right)·2^{\mathsf{\eta }}}}\right)+\left({\displaystyle \frac{\left(\mathsf{\eta }+5\right)·2^{\mathsf{\eta }}+3}{\left(6\mathsf{\eta }+6\right)·2^{\mathsf{\eta }}}}\right)\right)\parallel \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)|\otimes 1-1\otimes |{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\left|\right|\parallel ).\hfill \end{array}$$

(25)

□

5. Fractional Hermite–Hadamard Inequality Involving Arithmetic–Geometric Mean-Type Convexity

Lemma 5. 

Let the self-adjoint operators $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ have $\mathtt{Sp}\left(\mathsf{{\rm Y}}\right)\subset \mathsf{\Omega }$ and $\mathtt{Sp}\left(\mathsf{\zeta }\right)\subset$ Ω. Assume that $\mathsf{\vartheta }$ is a continuous and convex function on Ω; then, one has

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}-{\displaystyle \frac{2(\mathsf{\zeta }\otimes 1-\mathsf{{\rm Y}}\otimes 1)}{{(\sqrt{{\mathsf{\varpi }}_2}-\sqrt{{\mathsf{\varpi }}_1})}^2}}{\int }_0^1\mathsf{\vartheta }((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }\mathsf{\zeta }\otimes 1))\mathtt{d}\mathsf{\delta }\hfill \\ \hfill =& {\displaystyle \frac{{(\sqrt{1\otimes \mathsf{\zeta }}-\sqrt{\mathsf{{\rm Y}}\otimes 1})}^2}{4}}\left[{\int }_0^1\left((1-2\mathsf{\delta })\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }1\otimes \mathsf{\zeta }\right)\right]\mathtt{d}\mathsf{\delta }.\hfill \end{array}$$

(26)

Proof. 

Considering Lemma 4.1 from [82] based on differentiable convex mapping, one has

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)+\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)}{2}}-{\displaystyle \frac{2}{{(\sqrt{{\mathsf{\varpi }}_2}-\sqrt{{\mathsf{\varpi }}_1})}^2}}{\int }_{{\mathsf{\varpi }}_1}^{{\mathsf{\varpi }}_2}\mathsf{\vartheta }\left(\mathsf{\sigma }\right)\mathtt{d}\mathsf{\sigma }\hfill \\ & ={\displaystyle \frac{\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)+\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)}{2}}-{\displaystyle \frac{2({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}{{(\sqrt{{\mathsf{\varpi }}_2}-\sqrt{{\mathsf{\varpi }}_1})}^2}}{\int }_0^1\mathsf{\vartheta }((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2))\mathtt{d}\mathsf{\delta }\hfill \\ \hfill =& {\displaystyle \frac{{(\sqrt{{\mathsf{\varpi }}_2}-\sqrt{{\mathsf{\varpi }}_1})}^2}{4}}\left[{\int }_0^1(1-2\mathsf{\delta }){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2\right)\right]\mathtt{d}\mathsf{\delta }\hfill \end{array}$$

(27)

Assume that $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ have the spectral resolutions

$$\mathsf{{\rm Y}}={\int }_{\mathsf{\Omega }}{\mathsf{\varpi }}_2\mathtt{dE}\left({\mathsf{\varpi }}_2\right)\mathrm{and}\mathsf{\zeta }={\int }_{\mathsf{\Omega }}{\mathsf{\varpi }}_1\mathtt{dF}\left({\mathsf{\varpi }}_1\right).$$

If we take the integral ${\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}$ over ${\mathtt{dE}}_{{\mathsf{\varpi }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\varpi }}_2}$ in (27), then we obtain

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left[{\displaystyle \frac{\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)+\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)}{2}}\right]\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}-{\displaystyle \frac{2({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}{{(\sqrt{{\mathsf{\varpi }}_2}-\sqrt{{\mathsf{\varpi }}_1})}^2}}{\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}{\int }_0^1\mathsf{\vartheta }((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2))\mathtt{d}\mathsf{\delta }\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}\hfill \\ \hfill =& {\displaystyle \frac{{(\sqrt{{\mathsf{\varpi }}_2}-\sqrt{{\mathsf{\varpi }}_1})}^2}{4}}{\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left[{\int }_0^1(1-2\mathsf{\delta }){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2\right)\right]\mathtt{d}\mathsf{\delta }\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}\hfill \end{array}$$

(28)

Taking into account Lemma 1 and Fubini’s theorem, we obtain the following representations:

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\mathsf{\vartheta }\left({\mathsf{\varpi }}_2\right)\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}=(\mathsf{\vartheta }\left(\mathsf{\zeta }\right)\otimes 1),\hfill \\ \hfill & {\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}{\int }_0^1\mathsf{\vartheta }((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2))\mathtt{d}\mathsf{\delta }\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}={\int }_0^1\mathsf{\vartheta }((1-\mathsf{\delta })\mathsf{\zeta }\otimes 1+\mathsf{\delta }\otimes 1\mathsf{{\rm Y}}))\mathtt{d}\mathsf{\delta },\hfill \\ \hfill & {\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\mathsf{\vartheta }\left({\mathsf{\varpi }}_1\right)\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}=(1\otimes \mathsf{\vartheta }\left(\mathsf{{\rm Y}}\right)).\hfill \end{array}$$

(29)

A similar approach can be taken to the right side as well that is

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{(\sqrt{{\mathsf{\varpi }}_2}-\sqrt{{\mathsf{\varpi }}_1})}^2}{4}}{\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left[{\int }_0^1(1-2\mathsf{\delta }){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta }){\mathsf{\varpi }}_1+\mathsf{\delta }{\mathsf{\varpi }}_2\right)]\mathtt{d}\mathsf{\delta }\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}\right.\hfill \\ \hfill & ={\displaystyle \frac{{(\sqrt{1\otimes \mathsf{\zeta }}-\sqrt{\mathsf{{\rm Y}}\otimes 1})}^2}{4}}\left[{\int }_0^1\left((1-2\mathsf{\delta })\right){\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }1\otimes \mathsf{\zeta }\right)]\mathtt{d}\mathsf{\delta }\right.\hfill \end{array}$$

(30)

Using Equations (29) and (30) in (28), we obtain the required result. □

Theorem 10. 

Let the self-adjoint operators $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ have $\mathtt{Sp}\left(\mathsf{{\rm Y}}\right)\subset \mathsf{\Omega }$ and $\mathtt{Sp}\left(\mathsf{\zeta }\right)\subset$ Ω. Assume that $\mathsf{\vartheta }$ is differentiable on Ω with ${∥{\mathsf{\vartheta }}^{″}∥}_{\mathsf{\Omega },\infty }:=$ ${sup}_{\mathsf{\eta }\in \mathsf{\Omega }}\left|{\mathsf{\vartheta }}^{″}\left(\mathsf{\eta }\right)\right|<\infty$; then, one has

$$\begin{array}{cc}\hfill \parallel & {\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}-{\displaystyle \frac{2(\mathsf{\zeta }\otimes 1-\mathsf{{\rm Y}}\otimes 1)}{{(\sqrt{{\mathsf{\varpi }}_2}-\sqrt{{\mathsf{\varpi }}_1})}^2}}{\int }_0^1\mathsf{\vartheta }((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }\mathsf{\zeta }\otimes 1))\mathtt{d}\mathsf{\delta }\parallel \hfill \\ \hfill & \le {\displaystyle \frac{{(\sqrt{1\otimes \mathsf{\zeta }}-\sqrt{\mathsf{{\rm Y}}\otimes 1})}^2}{4}}{\displaystyle \frac{1}{1+\mathsf{\eta }}}\left(2-2^{1-\mathsf{\eta }}\right){∥{\mathsf{\vartheta }}^{\prime }∥}_{\mathsf{\Omega },\infty }.\hfill \end{array}$$

(31)

Proof. 

Using the triangle inequality and the operator norm of the previously derived Lemma 5, we may obtain

$$\begin{array}{cc}\hfill \parallel & {\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}-{\displaystyle \frac{2(\mathsf{\zeta }\otimes 1-\mathsf{{\rm Y}}\otimes 1)}{{(\sqrt{{\mathsf{\varpi }}_2}-\sqrt{{\mathsf{\varpi }}_1})}^2}}{\int }_0^1\mathsf{\vartheta }((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }\mathsf{\zeta }\otimes 1))\mathtt{d}\mathsf{\delta }\parallel \hfill \\ \hfill & \le {\displaystyle \frac{{(\sqrt{1\otimes \mathsf{\zeta }}-\sqrt{\mathsf{{\rm Y}}\otimes 1})}^2}{4}}{\int }_0^1∥\left((1-2\mathsf{\delta })\right)∥∥{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }1\otimes \mathsf{\zeta }\right.∥\mathtt{d}\mathsf{\delta }\hfill \end{array}$$

(32)

Considering Lemma 1, we obtain

$$\begin{array}{cc}\hfill & \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1+\left(1-\mathsf{\delta }\right)1\otimes \mathsf{\zeta }\right)\right|={\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left|\left({\mathsf{\vartheta }}^{\prime }\mathsf{\delta }{\mathsf{\varpi }}_1+\left(1-\mathsf{\delta }\right){\mathsf{\varpi }}_2\right)\right|\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}.\hfill \end{array}$$

Since

$$\left|\left({\mathsf{\vartheta }}^{\prime }(\mathsf{\delta }{\mathsf{\varpi }}_1+\left(1-\mathsf{\delta }\right){\mathsf{\varpi }}_2\right)\right|⩽{∥{\mathsf{\vartheta }}^{\prime }∥}_{\mathsf{\Omega },+\infty }$$

for all $\mathsf{\delta }\in [0,1]$ and ${\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2\in \mathsf{\Omega }$.

If we take the integral ${\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}$ over ${\mathtt{dE}}_{{\mathsf{\varpi }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\varpi }}_2}$, then we obtain

$$\begin{array}{cc}\hfill & \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }\mathsf{{\rm Y}}\otimes 1+\left(1-\mathsf{\delta }\right)1\otimes \mathsf{\zeta }\right)\right|={\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left|\left({\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }{\mathsf{\varpi }}_1\right)+\left(1-\mathsf{\delta }\right){\mathsf{\varpi }}_2\right)\right|\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}\hfill \\ \hfill & ⩽{∥{\mathsf{\vartheta }}^{\prime }∥}_{\mathsf{\Omega },+\infty }{\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}={∥{\mathsf{\vartheta }}^{\prime }∥}_{\mathsf{\Omega },+\infty }.\hfill \end{array}$$

(33)

From which we obtain the following:

$$\begin{array}{cc}\hfill & {\int }_0^1∥\left((1-2\mathsf{\delta })\right)∥∥{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }1\otimes \mathsf{\zeta }\right.∥\mathtt{d}\mathsf{\delta }\hfill \\ \hfill & ⩽{∥{\mathsf{\vartheta }}^{\prime }∥}_{\mathsf{\Omega },+\infty }{\int }_0^1\parallel ({(1-\mathsf{\delta })}^{\mathsf{\eta }}-{\mathsf{\delta }}^{\mathsf{\eta }})\parallel \mathtt{d}\mathsf{\delta }={\displaystyle \frac{1}{1+\mathsf{\eta }}}\left(2-2^{1-\mathsf{\eta }}\right){∥{\mathsf{\vartheta }}^{\prime }∥}_{\mathsf{\Omega },+\infty }.\hfill \end{array}$$

(34)

Using Equations (33) and (34) in (42), we obtain the required result. □

Theorem 11. 

Let the self-adjoint operators $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ have $\mathtt{Sp}\left(\mathsf{{\rm Y}}\right)\subset \mathsf{\Omega }$ and $\mathtt{Sp}\left(\mathsf{\zeta }\right)\subset$ Ω. Assume that $\mathsf{\vartheta }$ is differentiable on Ω and $\left|{\mathsf{\vartheta }}^{\prime }\right|$ is convex on Ω; then, one has

$$\begin{array}{cc}\hfill \parallel & {\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}-{\displaystyle \frac{2(\mathsf{\zeta }\otimes 1-\mathsf{{\rm Y}}\otimes 1)}{{(\sqrt{{\mathsf{\varpi }}_2}-\sqrt{{\mathsf{\varpi }}_1})}^2}}{\int }_0^1\mathsf{\vartheta }((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }\mathsf{\zeta }\otimes 1))\mathtt{d}\mathsf{\delta }\parallel \hfill \\ \hfill & \le {\displaystyle \frac{{(\sqrt{1\otimes \mathsf{\zeta }}-\sqrt{\mathsf{{\rm Y}}\otimes 1})}^2}{4}}{\displaystyle \frac{1}{1+\mathsf{\eta }}}\left(2-2^{1-\mathsf{\eta }}\right){∥{\mathsf{\vartheta }}^{\prime }∥}_{\mathsf{\Omega },\infty }.\hfill \end{array}$$

(35)

Proof. 

As $\left|{\mathsf{\vartheta }}^{\prime }\right|$ is convex on $\mathsf{\Omega }$, we obtain

$$\begin{array}{cc}\hfill & \left.\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }{\mathsf{\varpi }}_1\right)+\left(1-\mathsf{\delta }\right){\mathsf{\varpi }}_2\right)\right|\le \mathsf{\delta }\left|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_2\right)\right|+\left(1-\mathsf{\delta }\right)\left|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_1\right)\right|\hfill \end{array}$$

for all for $\mathsf{\delta }\in [0,1]$ and ${\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2\in \mathsf{\Omega }$.

If we take the integral ${\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}$ over ${\mathtt{dE}}_{{\mathsf{\varpi }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\varpi }}_2}$, then we obtain

$$\begin{array}{cc}\hfill & \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }\left(\mathsf{{\rm Y}}\otimes 1\right)+\left(1-\mathsf{\delta }\right)1\otimes \mathsf{\zeta }\right)\right|={\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left|\left({\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }\left({\mathsf{\varpi }}_1\right)+\left(1-\mathsf{\delta }\right){\mathsf{\varpi }}_2\right)\right|\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}\right.\hfill \\ \hfill & \le (1-\mathsf{\delta }){\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_1\right)\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}+\mathsf{\delta }{\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_2\right)\right|\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2},\hfill \end{array}$$

(36)

namely

$$\begin{array}{c}\hfill \left|{\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }1\otimes \mathsf{\zeta })\right|\le (1-\mathsf{\delta })\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+\mathsf{\delta }\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|\otimes 1\end{array}$$

(37)

for all $\mathsf{\delta }\in [0,1]$.

If we take the norm in (37), then we obtain

$$\begin{array}{cc}\hfill ∥{\mathsf{\vartheta }}^{\prime }((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }1\otimes \mathsf{\zeta })∥& \le ∥(1-\mathsf{\delta })\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+\mathsf{\delta }\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|\otimes 1∥\hfill \\ & \hfill \le (1-\mathsf{\delta })∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)∥+\mathsf{\delta }∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)∥\end{array}$$

(38)

Again using the triangle inequality and the operator norm of the previously derived Lemma 5, we may obtain

$$\begin{array}{cc}\hfill \parallel & {\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}-{\displaystyle \frac{2(\mathsf{\zeta }\otimes 1-\mathsf{{\rm Y}}\otimes 1)}{{(\sqrt{{\mathsf{\varpi }}_2}-\sqrt{{\mathsf{\varpi }}_1})}^2}}{\int }_0^1\mathsf{\vartheta }((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }\mathsf{\zeta }\otimes 1))\mathtt{d}\mathsf{\delta }\parallel \hfill \\ \hfill & \le {\displaystyle \frac{{(\sqrt{1\otimes \mathsf{\zeta }}-\sqrt{\mathsf{{\rm Y}}\otimes 1})}^2}{4}}{\int }_0^1∥\left((1-2\mathsf{\delta })\right)∥∥{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }1\otimes \mathsf{\zeta }\right.∥\mathtt{d}\mathsf{\delta }\hfill \\ \hfill & \le {\displaystyle \frac{{(\sqrt{1\otimes \mathsf{\zeta }}-\sqrt{\mathsf{{\rm Y}}\otimes 1})}^2}{4}}{\int }_0^1∥(1-2\mathsf{\delta })∥\left(\mathsf{\delta }∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)∥+(1-\mathsf{\delta })∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)∥\right)\mathtt{d}\mathsf{\delta }\hfill \\ \hfill =& {\displaystyle \frac{{(\sqrt{1\otimes \mathsf{\zeta }}-\sqrt{\mathsf{{\rm Y}}\otimes 1})}^2}{4}}{\displaystyle \frac{\left(∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)∥+∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)∥\right)}{2}}.\hfill \end{array}$$

(39)

□

Theorem 12. 

Let the self-adjoint operators $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ have $\mathtt{Sp}\left(\mathsf{{\rm Y}}\right)\subset \mathsf{\Omega }$ and $\mathtt{Sp}\left(\mathsf{\zeta }\right)\subset$ Ω. Assume that $\mathsf{\vartheta }$ is continuously differentiable on Ω with $\left|{\mathsf{\vartheta }}^{\prime }\right|$ being quasi-convex on Ω; then, one has

$$\begin{array}{cc}\hfill \parallel & {\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}-{\displaystyle \frac{2(\mathsf{\zeta }\otimes 1-\mathsf{{\rm Y}}\otimes 1)}{{(\sqrt{{\mathsf{\varpi }}_2}-\sqrt{{\mathsf{\varpi }}_1})}^2}}{\int }_0^1\mathsf{\vartheta }((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }\mathsf{\zeta }\otimes 1))\mathtt{d}\mathsf{\delta }\parallel \hfill \\ \hfill & \le {\displaystyle \frac{{(\sqrt{1\otimes \mathsf{\zeta }}-\sqrt{\mathsf{{\rm Y}}\otimes 1})}^2}{4}}{\displaystyle \frac{1}{1+\mathsf{\eta }}}\left(2-2^{1-\mathsf{\eta }}\right){∥{\mathsf{\vartheta }}^{\prime }∥}_{\mathsf{\Omega },\infty }.\hfill \end{array}$$

(40)

Proof. 

As $\left|{\mathsf{\vartheta }}^{\prime }\right|$ is quasi-convex on $\mathsf{\Omega }$, one has

$$\begin{array}{cc}\hfill & \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }\left({\mathsf{\varpi }}_1\right)+\left(1-\mathsf{\delta }\right){\mathsf{\varpi }}_2\right)\right|\le {\displaystyle \frac{1}{2}}\left(\left|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_2\right)\right|+\left|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_1\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_2\right)|-|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_1\right)\left|\right|\right)\hfill \end{array}$$

for all for $\mathsf{\delta }\in [0,1]$ and ${\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2\in \mathsf{\Omega }$.

Taking the double integral ${\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}$ over ${\mathtt{dE}}_{{\mathsf{\varpi }}_1}\otimes {\mathtt{dF}}_{{\mathsf{\varpi }}_2}$ yields

$$\begin{array}{cc}\hfill & \left|\left({\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }\left(\mathsf{{\rm Y}}\otimes 1\right)+\left(1-\mathsf{\delta }\right)1\otimes \mathsf{\zeta }\right)\right|\right.\hfill \\ \hfill & ={\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left|\left({\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }\left({\mathsf{\varpi }}_2\right)+\left(1-\mathsf{\delta }\right){\mathsf{\varpi }}_2\right)\right|\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}\right.\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}\left(\left|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_2\right)\right|+\left|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_1\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_2\right)|-|{\mathsf{\vartheta }}^{\prime }\left({\mathsf{\varpi }}_1\right)\left|\right|\right)\mathtt{d}{\mathtt{E}}_{{\mathsf{\varpi }}_1}\otimes \mathtt{d}{\mathtt{F}}_{{\mathsf{\varpi }}_2}\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}\left(\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)|\otimes 1-1\otimes |{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\left|\right|\right).\hfill \end{array}$$

for all $\mathsf{\delta }\in [0,1]$.

If we take the norm in the inequality above, we obtain the following:

$$\begin{array}{cc}\hfill & ∥\left({\mathsf{\vartheta }}^{\prime }\left(\mathsf{\delta }\left(\mathsf{{\rm Y}}\otimes 1\right)+\left(1-\mathsf{\delta }\right)1\otimes \mathsf{\zeta }\right)\right.∥\hfill \\ \hfill & \le ∥{\displaystyle \frac{1}{2}}\left(\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)|\otimes 1-1\otimes |{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\left|\right|\right)∥\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{2}}\left(∥\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|∥+∥\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1-1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|∥\right).\hfill \end{array}$$

Using the triangle inequality and the operator norm of the previously derived Lemma 5, we may obtain

$$\begin{array}{cc}\hfill \parallel & {\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}-{\displaystyle \frac{2(\mathsf{\zeta }\otimes 1-\mathsf{{\rm Y}}\otimes 1)}{{(\sqrt{{\mathsf{\varpi }}_2}-\sqrt{{\mathsf{\varpi }}_1})}^2}}{\int }_0^1\mathsf{\vartheta }((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }\mathsf{\zeta }\otimes 1))\mathtt{d}\mathsf{\delta }\parallel \hfill \\ \hfill & \le {\displaystyle \frac{{(\sqrt{1\otimes \mathsf{\zeta }}-\sqrt{\mathsf{{\rm Y}}\otimes 1})}^2}{4}}{\int }_0^1∥\left((1-2\mathsf{\delta })\right)∥∥{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\mathsf{\delta }1\otimes \mathsf{\zeta }\right.∥\mathtt{d}\mathsf{\delta }\hfill \\ \hfill & \le {\displaystyle \frac{{(\sqrt{1\otimes \mathsf{\zeta }}-\sqrt{\mathsf{{\rm Y}}\otimes 1})}^2}{4}}{\int }_0^1∥(1-2\mathsf{\delta })∥\left(\mathsf{\delta }∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)∥+(1-\mathsf{\delta })∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)∥\right)\mathtt{d}\mathsf{\delta }\hfill \\ \hfill & \le {\displaystyle \frac{{(\sqrt{1\otimes \mathsf{\zeta }}-\sqrt{\mathsf{{\rm Y}}\otimes 1})}^2}{4}}{\int }_0^1∥(1-2\mathsf{\delta })∥∥{\displaystyle \frac{1}{2}}\left(\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|+\left|\right|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)|\otimes 1-1\otimes |{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\left|\right|\right)∥\hfill \\ \hfill & ={\displaystyle \frac{{(\sqrt{1\otimes \mathsf{\zeta }}-\sqrt{\mathsf{{\rm Y}}\otimes 1})}^2}{4}}{\displaystyle \frac{1}{1+\mathsf{\eta }}}\left(2-2^{1-\mathsf{\eta }}\right){\displaystyle \frac{1}{2}}\left(∥\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1+1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|∥+∥\left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\right|\otimes 1-1\otimes \left|{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\right|∥\right)\hfill \end{array}$$

(41)

□

6. Some Examples and Consequences

The following property is satisfied by the exponential function if $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ are commuting; that is, if $\mathsf{{\rm Y}}\mathsf{\zeta }=\mathsf{\zeta }\mathsf{{\rm Y}}$, then one has

$$e^{\mathsf{{\rm Y}}}{e}^{\mathsf{\zeta }}=e^{\mathsf{\zeta }}{e}^{\mathsf{{\rm Y}}}=e^{(\mathtt{G}+\mathtt{H})}.$$

Also, if $\mathsf{{\rm Y}}$ is invertible and ${\mathsf{\varpi }}_1,{\mathsf{\varpi }}_2\in \mathsf{R}$ with ${\mathsf{\varpi }}_1<{\mathsf{\varpi }}_2$, then

$${\int }_{{\mathsf{\varpi }}_1}^{{\mathsf{\varpi }}_2}{e}^{\mathsf{\delta }\mathsf{{\rm Y}}}\mathtt{d}\mathsf{\delta }={\displaystyle \frac{[e^{{\mathsf{\varpi }}_2\mathsf{{\rm Y}}}-e^{{\mathsf{\varpi }}_1\mathsf{{\rm Y}}}]}{\mathsf{{\rm Y}}}}.$$

Moreover, if $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ are commuting and $\mathsf{\zeta }-\mathsf{{\rm Y}}$ is invertible, then

$$\begin{array}{cc}\hfill {\int }_0^1{e}^{((1-{\mathsf{\varpi }}_2)\mathsf{{\rm Y}}+\mathsf{\eta }\mathsf{\zeta })}\mathtt{d}\mathsf{\eta }& ={\int }_0^1{e}^{\left(\mathsf{\eta }\right(\mathsf{\zeta }-\mathsf{{\rm Y}}\left)\right)}{e}^{\mathsf{{\rm Y}}}\mathtt{d}\mathsf{\eta }=\left({\int }_0^1{e}^{\left(\mathsf{\eta }\right(\mathsf{\zeta }-\mathsf{{\rm Y}}\left)\right)}\mathtt{d}\mathsf{\eta }\right)e^{\mathsf{{\rm Y}}}\hfill \\ & ={\displaystyle \frac{[e^{(\mathsf{\zeta }-\mathsf{{\rm Y}})}-\mathtt{I}]e^{\mathsf{{\rm Y}}}}{\mathsf{\zeta }-\mathsf{{\rm Y}}}}={\displaystyle \frac{[e^{\mathsf{\zeta }}-e^{\mathsf{{\rm Y}}}]}{\mathsf{\zeta }-\mathsf{{\rm Y}}}}.\hfill \end{array}$$

Given that the operators $\mathsf{{\rm Y}}=\mathtt{L}\otimes 1$ and $\mathsf{\zeta }=1\otimes \mathtt{M}$ are commutative, and $1\otimes \mathtt{M}-\mathtt{L}\otimes 1$ is invertible, then

$$\begin{array}{cc}& {\int }_0^1{e}^{\left(\right(1-\mathsf{\eta })\mathtt{L}\otimes 1+\mathsf{\eta }1\otimes \mathtt{M})}\mathtt{d}\mathsf{\eta }={\displaystyle \frac{[e^{(1\otimes \mathtt{M})}-e^{(\mathtt{L}\otimes 1)}]}{(1\otimes \mathtt{M}-\mathtt{L}\otimes 1)}}\hfill \end{array}$$

Corollary 2. 

Let the self-adjoint operators $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ have $\mathtt{Sp}\left(\mathsf{{\rm Y}}\right)\subset \mathsf{\Omega }$ and $\mathtt{Sp}\left(\mathsf{\zeta }\right)\subset$ Ω. Assume that $\mathsf{\vartheta }$ is differentiable on $\mathsf{\Omega },\mathsf{\eta }={\displaystyle \frac{1}{4}}$ with ${∥{\mathsf{\vartheta }}^{″}∥}_{\mathsf{\Omega },\infty }:=$ ${sup}_{\mathsf{\eta }\in \mathsf{\Omega }}\left|{\mathsf{\vartheta }}^{″}\left(\mathsf{\eta }\right)\right|<\infty$; then, by Theorem 7, we have

$$\begin{array}{cc}\hfill & \parallel \left({\displaystyle \frac{2}{3}}(exp\left(\mathsf{{\rm Y}}\right)\otimes 1)-{\displaystyle \frac{1}{3}}exp\left({\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}\right)+{\displaystyle \frac{2}{3}}(1\otimes \mathsf{\vartheta }\left(\mathsf{\zeta }\right))\right)-{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{5}{4}\right)}{{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\frac{1}{4}}}}\left({\displaystyle \frac{1}{2^{\frac{1}{4}-1}\mathsf{\Gamma }\left(\frac{5}{4}\right)}}{\displaystyle \frac{exp\left(\mathsf{{\rm Y}}\right)+\mathsf{\vartheta }\left(\mathsf{\zeta }\right)}{2}}\right.\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\frac{5}{4}\right)}}\left({\displaystyle \frac{{(1-\mathsf{\delta })}^{\frac{1}{4}}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\left(\frac{5}{4}\right)}}{2^{\left(\frac{5}{4}\right)}\mathsf{\Gamma }\left(\frac{5}{4}\right)}}\left({\int }_0^1{exp}^{\prime }\left((1-\mathsf{\delta })\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+\mathsf{\zeta }\otimes 1}{2}}\right)+1\otimes \mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right)\right.\hfill \\ \hfill & +{\displaystyle \frac{{\mathsf{\delta }}^{\frac{1}{4}}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\left(\frac{5}{4}\right)}}{2^{\left(\frac{5}{4}\right)}\mathsf{\Gamma }\left(\frac{5}{4}\right)}}\left({\int }_0^1{exp}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+1\otimes \mathsf{\zeta }}{2}}\right)\mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right))\parallel \hfill \\ & \le {\displaystyle \frac{\parallel 1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1\parallel }{2}}\left(2{∥{exp}^{\prime }∥}_{\mathsf{\Omega },+\infty }\left({\displaystyle \frac{\left(\frac{5}{2}\right)·2^{\frac{1}{4}}-3}{\left(9\right)·2^{\frac{1}{4}}}}\right)+2{∥{exp}^{\prime }∥}_{\mathsf{\Omega },+\infty }\left({\displaystyle \frac{\left(\frac{21}{4}\right)·2^{\frac{1}{4}}+3}{\left(9\right)·2^{\frac{1}{4}}}}\right)\right).\hfill \end{array}$$

Corollary 3. 

Let the self-adjoint operators $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ have $\mathtt{Sp}\left(\mathsf{{\rm Y}}\right)\subset \mathsf{\Omega }$ and $\mathtt{Sp}\left(\mathsf{\zeta }\right)\subset$ Ω. Assume that $\mathsf{\vartheta }$ is differentiable on Ω and $\left|{\mathsf{\vartheta }}^{\prime }\right|$ is convex on $\mathsf{\Omega },\mathsf{\eta }={\displaystyle \frac{1}{2}}$; then, by Theorem 8, we have

$$\begin{array}{cc}\hfill & \parallel \left({\displaystyle \frac{2}{3}}(ln\left(\mathsf{{\rm Y}}\right)\otimes 1)-{\displaystyle \frac{1}{3}}ln\left({\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}\right)+{\displaystyle \frac{2}{3}}(1\otimes \mathsf{\vartheta }\left(\mathsf{\zeta }\right))\right)-{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{5}{4}\right)}{{({\mathsf{\varpi }}_2-{\mathsf{\varpi }}_1)}^{\frac{1}{4}}}}\left({\displaystyle \frac{1}{2^{\frac{1}{4}-1}\mathsf{\Gamma }\left(\frac{5}{4}\right)}}{\displaystyle \frac{ln\left(\mathsf{{\rm Y}}\right)+\mathsf{\vartheta }\left(\mathsf{\zeta }\right)}{2}}\right.\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\frac{5}{4}\right)}}\left({\displaystyle \frac{{(1-\mathsf{\delta })}^{\frac{1}{4}}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\left(\frac{5}{4}\right)}}{2^{\left(\frac{5}{4}\right)}\mathsf{\Gamma }\left(\frac{5}{4}\right)}}\left({\int }_0^1{ln}^{\prime }\left((1-\mathsf{\delta })\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+\mathsf{\zeta }\otimes 1}{2}}\right)+1\otimes \mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right)\right.\hfill \\ \hfill & +{\displaystyle \frac{{\mathsf{\delta }}^{\frac{1}{4}}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\left(\frac{5}{4}\right)}}{2^{\left(\frac{5}{4}\right)}\mathsf{\Gamma }\left(\frac{5}{4}\right)}}\left({\int }_0^1{ln}^{\prime }\left((1-\mathsf{\delta })1\otimes \mathsf{\zeta }+\left({\displaystyle \frac{1\otimes \mathsf{{\rm Y}}+1\otimes \mathsf{\zeta }}{2}}\right)\mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right))\parallel \hfill \\ \hfill & \le {\displaystyle \frac{\parallel 1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1\parallel }{2}}\left(\left({\displaystyle \frac{\frac{9\left(\sqrt{2}\right)+24}{\left(180\right)\sqrt{2}}}{45}}\right)∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)\otimes 1∥+\left({\displaystyle \frac{24·2^{-\frac{5}{2}}-3·2^{-\frac{1}{2}}+\frac{5}{2}-14}{30}}\right)∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)\otimes 1∥\right).\hfill \end{array}$$

Corollary 4. 

Let the self-adjoint operators $\mathsf{{\rm Y}}$ and $\mathsf{\zeta }$ have $\mathtt{Sp}\left(\mathsf{{\rm Y}}\right)\subset \mathsf{\Omega }$ and $\mathtt{Sp}\left(\mathsf{\zeta }\right)\subset$ Ω. Assume that $\mathsf{\vartheta }$ is differentiable on Ω and $\left|{\mathsf{\vartheta }}^{\prime }\right|$ is convex on $\mathsf{\Omega },\mathsf{\eta }={\displaystyle \frac{1}{3}}$; then, by Theorem 11, we have

$$\begin{array}{cc}\hfill \parallel & {\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}-{\displaystyle \frac{2^{\frac{-2}{3}}\mathsf{\Gamma }\left(\frac{4}{3}\right)}{{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\frac{1}{3}}}}\left[{\displaystyle \frac{1}{2^{\frac{-2}{3}}\mathsf{\Gamma }\left(\frac{4}{3}\right)}}{\displaystyle \frac{\mathsf{{\rm Y}}\otimes 1+1\otimes \mathsf{\zeta }}{2}}\right.\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\frac{4}{3}\right)}}\left[{\displaystyle \frac{{(1-\mathsf{\delta })}^{\frac{1}{3}}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\frac{4}{3}}}{2^{\frac{4}{3}}\mathsf{\Gamma }\left(\frac{4}{3}\right)}}\left[{\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\left({\displaystyle \frac{1\otimes \mathsf{\zeta }+\mathsf{{\rm Y}}\otimes 1}{2}}\right)+\mathsf{\zeta }\otimes 1\mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right]\right.\hfill \\ \hfill & \left.\left.+{\displaystyle \frac{{\mathsf{\delta }}^{\frac{1}{3}}{(1\otimes \mathsf{\zeta }-\mathsf{{\rm Y}}\otimes 1)}^{\frac{4}{3}}}{2^{\frac{4}{3}}\mathsf{\Gamma }\left(\frac{4}{3}\right)}}\left[{\int }_0^1{\mathsf{\vartheta }}^{\prime }\left((1-\mathsf{\delta })\mathsf{{\rm Y}}\otimes 1+\left({\displaystyle \frac{1\otimes \mathsf{\zeta }+\mathsf{{\rm Y}}\otimes 1}{2}}\right)\mathsf{\delta }\right)\mathtt{d}\mathsf{\delta }\right]\right]\right]\parallel \hfill \\ & \le {\displaystyle \frac{{(\sqrt{1\otimes \mathsf{\zeta }}-\sqrt{\mathsf{{\rm Y}}\otimes 1})}^2}{4}}{\displaystyle \frac{2^{-\frac{1}{3}}\left(2^{\frac{1}{3}}-1\right)\left(∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{\zeta }\right)∥+∥{\mathsf{\vartheta }}^{\prime }\left(\mathsf{{\rm Y}}\right)∥\right)}{\frac{4}{3}}}.\hfill \end{array}$$

(42)

7. Hermite–Hadamard Inequality in Mixed-Norm Morrey Spaces with Variable Exponent

In this section, we develop a double inequality for mixed-norm Moore spaces with variable exponents, based on the results of [83,84] derived from classical Lebesgue spaces, which we refine and generalize by applying a new Definition 6 with very interesting exponent settings.

Theorem 13. 

Suppose $\mathtt{p},\mathtt{q},\mathtt{v}\in \mathtt{P}\left({\mathsf{R}}^{\mathtt{n}}\right)$ such that $1\le \mathtt{q}\left(\mathsf{\mu }\right)\le \mathtt{p}\left(\mathsf{\mu }\right)\le \mathtt{v}\left(\mathsf{\mu }\right)\le \infty$ almost everywhere ${\displaystyle \frac{1}{\mathtt{p}\left(\mathsf{\mu }\right)}}+{\displaystyle \frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}\le 1$ and for each $\mathsf{\mu }\in {\mathsf{R}}^{\mathtt{n}}$ with $\mathsf{\Omega }=[0,1]$; then, one has

$${∥{\displaystyle \frac{{\xi }_{\mathsf{\eta }}+{\mathsf{\vartheta }}_{\mathsf{\eta }}}{2}}∥}_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}(·),\mathtt{v}(·)}\right)}\le {\int }_{\mathsf{\Omega }}{∥(1-\mathsf{\sigma }){\xi }_{\mathsf{\eta }}+\mathsf{\sigma }{\mathsf{\vartheta }}_{\mathsf{\eta }}∥}_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}(·),\mathtt{v}(·)}\right)}\mathtt{d}\mathsf{\sigma }\le {\displaystyle \frac{\parallel {\xi }_{\mathsf{\eta }}{\parallel }_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}(·),\mathtt{v}(·)}\right)}+{\parallel {\mathsf{\vartheta }}_{\mathsf{\eta }}\parallel }_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}(·),\mathtt{v}(·)}\right)}}{2}}.$$

Proof. 

If $1\le \mathtt{p}\left(\mathsf{\mu }\right)\le \mathtt{v}\left(\mathsf{\mu }\right)$ and $1\le \mathtt{q}$ are constant, then the proof is trivial. In the remaining situations, we would like to demonstrate that

$$\begin{array}{c}\hfill {∥{\displaystyle \frac{{\xi }_{\mathsf{\eta }}+{\mathsf{\vartheta }}_{\mathsf{\eta }}}{2}}∥}_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}(·),\mathtt{v}(·)}\right)}\le {\displaystyle \frac{\parallel {\xi }_{\mathsf{\eta }}{\parallel }_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}(·),\mathtt{v}(·)}\right)}+{\parallel {\mathsf{\vartheta }}_{\mathsf{\eta }}\parallel }_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}(·),\mathtt{v}(·)}\right)}}{2}}\end{array}$$

(43)

for all measurable function sequences ${\left({\xi }_{\mathsf{\eta }}\right)}_{\mathsf{\eta }=1}^{\infty }$ and ${\left({\mathsf{\vartheta }}_{\mathsf{\eta }}\right)}_{\mathsf{\eta }=1}^{\infty }$. Let $2{\mathsf{\eta }}_1>0$ and $2{\mathsf{\eta }}_2>0$ be given with

$$\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\varrho }_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}}(·)\right)}\left({\displaystyle \frac{{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\xi }_{\mathsf{\eta }}{\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}}{2{\mathsf{\eta }}_1}}\right)\le 1\mathrm{and}\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\varrho }_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}}(·)\right)}\left({\displaystyle \frac{{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\mathsf{\vartheta }}_{\mathsf{\eta }}{\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}}{2{\mathsf{\eta }}_2}}\right)\le 1.$$

Now, we want to show that

$$\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\varrho }_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}}(·)\right)}\left({\displaystyle \frac{{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}({\xi }_{\mathsf{\eta }}+{\mathsf{\vartheta }}_{\mathsf{\eta }}){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}}{2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2}}\right)\le 1.$$

For each $ϵ>0$, we have a sequence of positive numbers ${\left\{{\varsigma }_{\mathsf{\eta }}\right\}}_{\mathsf{\eta }=1}^{\infty }$ and ${\left\{{\kappa }_{\mathsf{\eta }}\right\}}_{\mathsf{\eta }=1}^{\infty }$ such that

$$\begin{array}{c}\hfill \underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\varrho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}}{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)\le 1\mathrm{and}\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\varrho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}}{2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)\le 1,\end{array}$$

(44)

with

$$\sum _{\mathsf{\eta }=0}^{\infty }{\varsigma }_{\mathsf{\eta }}\le 1+{\displaystyle \frac{ϵ}{2}}\mathrm{also}\sum _{\mathsf{\eta }=0}^{\infty }{\kappa }_{\mathsf{\eta }}\le 1+{\displaystyle \frac{ϵ}{2}}$$

We define

$${\mathtt{A}}_{\mathsf{\eta }}:={\displaystyle \frac{{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}+{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}}{2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2}},\mathrm{that}\mathrm{is},\sum _{\mathsf{\eta }=0}^{\infty }{\mathtt{A}}_{\mathsf{\eta }}\le 1+{\displaystyle \frac{ϵ}{2}}$$

We will now show that

$$\begin{array}{c}\hfill \underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\varrho }_{p(·)}\left({\displaystyle \frac{{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}({\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right)+{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right)){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}}{{\left(\frac{{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}+{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}}{2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2}\right)}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}\left(2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2\right)}}\right)\le 1\forall \mathsf{\eta }\in \mathsf{N}.\end{array}$$

(45)

Let ${\mathsf{\Theta }}_0:=\left\{\mathsf{\mu }\in {\mathsf{R}}^{\mathtt{n}}:\mathtt{p}\left(\mathsf{\mu }\right)<\infty \right\}$ and ${\mathsf{\Theta }}_{\infty }:=\left\{\mathsf{\mu }\in {\mathsf{R}}^{\mathtt{n}}:\mathtt{p}\left(\mathsf{\mu }\right)=\infty \right\}$. Consider for each $\mathsf{\mu }\in {\mathsf{\Theta }}_0$

$${\mathsf{\zeta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right):=\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\left({\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\mathrm{and}{\mathtt{J}}_{\mathsf{\eta }}\left(\mathsf{\mu }\right):=\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\left({\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}.$$

Then, (44) may be reformulated as

$$\begin{array}{c}\hfill {\int }_{{\mathsf{\Theta }}_0}{\mathsf{\zeta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right)\mathtt{d}\mathsf{\mu }=\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\int }_{{\mathsf{\Theta }}_0}{\left({\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\mathtt{d}\mathsf{\mu }\le 1\mathrm{and}\underset{\mathsf{\mu }\in {\mathsf{\Theta }}_{\infty },x\in {\mathbb{R}}^n,r>0}{ess-sup}{\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\le 1\end{array}$$

(46)

and

$$\begin{array}{c}\hfill \underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\int }_{{\mathsf{\Theta }}_0}{\left({\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\mathtt{d}\mathsf{\mu }\le 1\mathrm{and}\underset{\mathsf{\mu }\in {\mathsf{\Theta }}_{\infty },x\in {\mathbb{R}}^n,r>0}{ess-sup}{\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\le 1.\end{array}$$

(47)

Next, we aim to confirm (45); that is,

$$\begin{array}{c}\hfill \underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\int }_{{\mathsf{\Theta }}_0}{\left({\displaystyle \frac{{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}\left(\left|{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right)+{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right)\right|\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}}{{\left(\frac{{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}+{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}}{2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2}\right)}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}\left(2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2\right)}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\mathtt{d}\mathsf{\mu }\le 1and\underset{\mathsf{\mu }\in {\mathsf{\Theta }}_{\infty },x\in {\mathbb{R}}^n,r>0}{ess-sup}{\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}({\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right)+{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right)){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{{\left(\frac{{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}+{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}}{2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2}\right)}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}\left(2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2\right)}}\le 1.\end{array}$$

(48)

First, we set up the second part of (48). Additionally, we see that (46) and (47) gives that

$$\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2}}\le {\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}^{{\displaystyle \frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}\mathrm{and}\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2}}\le {\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}^{{\displaystyle \frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}$$

hold for almost every $\mathsf{\mu }\in {\mathsf{\Theta }}_{\infty }$. Using $1\le \mathtt{q}\left(\mathsf{\mu }\right)$ and Hölder’s inequality in the form

$${\displaystyle \frac{{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}+{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}{2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2}}\le {\left({\displaystyle \frac{{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}+{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}}{2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2}}\right)}^{{\displaystyle \frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}},$$

we obtain that

$$\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\displaystyle \frac{{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}\left|{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right)+{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right)\right|{\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}}{{\left(\frac{{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}+{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}}{2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2}\right)}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}\left(2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2\right)}}\le 1.$$

We will now verify the first part of (48). Let $1\le \mathtt{q}\left(\mathsf{\mu }\right)\le \mathtt{p}\left(\mathsf{\mu }\right)<\infty$ for each $\mathsf{\mu }\in {\mathsf{\Theta }}_0$. Following that, we use Hölder’s inequality as follows:

$$\begin{array}{cc}\hfill & {\left(\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\left({\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\right)}^{{\displaystyle \frac{1}{\mathtt{p}\left(\mathsf{\mu }\right)}}}{\varsigma }_{\mathsf{\eta }}^{{\displaystyle \frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}2{\mathsf{\eta }}_1+{\left(\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\left({\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\right)}^{{\displaystyle \frac{1}{\mathtt{p}\left(\mathsf{\mu }\right)}}}{\kappa }_{\mathsf{\eta }}^{{\displaystyle \frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}2{\mathsf{\eta }}_2\hfill \\ \hfill & \le {\left(2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2\right)}^{1-{\displaystyle \frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}{\left(2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}+2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}\right)}^{{\displaystyle \frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}-{\displaystyle \frac{1}{\mathtt{p}\left(\mathsf{\mu }\right)}}}\hfill \\ \hfill & \times {\left(\left(\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\left({\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\right){\varsigma }_{\mathsf{\eta }}2{\mathsf{\eta }}_1+\left(\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\left({\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\right){\kappa }_{\mathsf{\eta }}2{\mathsf{\eta }}_2\right)}^{{\displaystyle \frac{1}{\mathtt{p}\left(\mathsf{\mu }\right)}}}\hfill \end{array}$$

(49)

If ${\displaystyle \frac{1}{\mathtt{p}\left(\mathsf{\mu }\right)}}+{\displaystyle \frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}\le 1$ almost everywhere and for each $\mathsf{\mu }\in {\mathsf{\Theta }}_0$, then we replace (49) with

$$\begin{array}{cc}& {\left(\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\left({\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\right)}^{{\displaystyle \frac{1}{\mathtt{p}\left(\mathsf{\mu }\right)}}}{\varsigma }_{\mathsf{\eta }}^{{\displaystyle \frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}2{\mathsf{\eta }}_1+{\left(\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\left({\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\right)}^{{\displaystyle \frac{1}{\mathtt{p}\left(\mathsf{\mu }\right)}}}{\kappa }_{\mathsf{\eta }}^{{\displaystyle \frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}2{\mathsf{\eta }}_2\hfill \\ \hfill & \le {\left(2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2\right)}^{1-{\displaystyle \frac{1}{\mathtt{p}\left(\mathsf{\mu }\right)}}-{\displaystyle \frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}{\left(2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}+2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}\right)}^{{\displaystyle \frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}\left(\left(\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\left({\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\right){\varsigma }_{\mathsf{\eta }}2{\mathsf{\eta }}_1\right.\hfill \\ & {\left.+\left(\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\left({\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\right){\kappa }_{\mathsf{\eta }}2{\mathsf{\eta }}_2\right)}^{{\displaystyle \frac{1}{\mathtt{p}\left(\mathsf{\mu }\right)}}}.\hfill \end{array}$$

(50)

For the sake of simplicity, we may consider

$$\begin{array}{c}\hfill F_v={\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}{G}_v={\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}},\end{array}$$

(51)

and by using (49), we may proceed as follows:

$$\begin{array}{cc}\hfill & \underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\int }_{{\mathsf{\Theta }}_0}{\left({\displaystyle \frac{{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}\left|{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right)+{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right)\right|{\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}}{{\left(\frac{{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}+{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}}{2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2}\right)}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}\left(2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2\right)}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\mathtt{d}\mathsf{\mu }\hfill \\ \hfill & \le {\int }_{{\mathsf{\Theta }}_0}{\left({\displaystyle \frac{{\left({sup}_{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{\left(F_v\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\right)}^{\frac{1}{\mathtt{p}\left(\mathsf{\mu }\right)}}{\varsigma }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}2{\mathsf{\eta }}_1+{\left({sup}_{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{\left(G_v\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\right)}^{\frac{1}{\mathtt{p}\left(\mathsf{\mu }\right)}}{\kappa }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}2{\mathsf{\eta }}_2}{2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}·{\left({\displaystyle \frac{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}+2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}}{2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2}}\right)}^{-{\displaystyle \frac{\mathtt{p}\left(\mathsf{\mu }\right)}{\mathtt{q}\left(\mathsf{\mu }\right)}}}\mathtt{d}\mathsf{\mu }\hfill \\ \hfill & \le {\int }_{{\mathsf{\Theta }}_0}{\displaystyle \frac{\left({sup}_{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{\left(\frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\right){\varsigma }_{\mathsf{\eta }}2{\mathsf{\eta }}_1+\left({sup}_{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{\left(\frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\right){\kappa }_{\mathsf{\eta }}2{\mathsf{\eta }}_2}{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}+2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}}}\mathtt{d}\mathsf{\mu }\hfill \\ \hfill & \le {\displaystyle \frac{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}}{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}+2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}}}{\int }_{{\mathsf{\Theta }}_0}\left(\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\left({\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\right)\mathtt{d}\mathsf{\mu }\hfill \\ & +{\displaystyle \frac{2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}}{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}+2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}}}{\int }_{{\mathsf{\Theta }}_0}\left(\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\left({\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\right)\mathtt{d}\mathsf{\mu }\le 1,\hfill \end{array}$$

(52)

where we also used (46) and (47). Alternatively, starting with (52), we proceed as follows:

$$\begin{array}{cc}\hfill & \underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\int }_{{\mathsf{\Theta }}_0}{\left({\displaystyle \frac{{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}\left|{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right)+{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{{\left(\frac{{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}+{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}}{2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2}\right)}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}\left(2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2\right)}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\mathtt{d}\mathsf{\mu }\hfill \\ \hfill & ={\int }_{{\mathsf{\Theta }}_0}{\left({\displaystyle \frac{{\left({sup}_{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{\left(F_v\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\right)}^{\frac{1}{\mathtt{p}\left(\mathsf{\mu }\right)}}{\varsigma }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}2{\mathsf{\eta }}_1+{\left(G_v\right)}^{\frac{1}{\mathtt{p}\left(\mathsf{\mu }\right)}}{\kappa }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}2{\mathsf{\eta }}_2}{2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}·{\left({\displaystyle \frac{{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}+{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}}{{\mathsf{\eta }}_1+{\mathsf{\eta }}_2}}\right)}^{-{\displaystyle \frac{\mathtt{p}\left(\mathsf{\mu }\right)}{\mathtt{q}\left(\mathsf{\mu }\right)}}}\mathtt{d}\mathsf{\mu }\hfill \\ \hfill & \le {\int }_{{\mathsf{\Theta }}_0}{\displaystyle \frac{\left({sup}_{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{\left(\frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\right)2{\mathsf{\eta }}_1+\left({sup}_{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{\left(\frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\right)2{\mathsf{\eta }}_2}{2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2}}\mathtt{d}\mathsf{\mu }\hfill \\ \hfill & ={\displaystyle \frac{2{\mathsf{\eta }}_1}{2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2}}\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\int }_{{\mathsf{\Theta }}_0}{\left({\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\xi }_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_1{\varsigma }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\mathtt{d}\mathsf{\mu }+{\displaystyle \frac{2{\mathsf{\eta }}_2}{2{\mathsf{\eta }}_1+2{\mathsf{\eta }}_2}}\underset{\sigma \in {\mathsf{R}}^{\mathtt{n}},\mathtt{r}>0}{sup}{\int }_{{\mathsf{\Theta }}_0}{\left({\displaystyle \frac{\left|{\mathtt{r}}^{\frac{\mathtt{n}}{\mathtt{v}\left(\sigma \right)}-\frac{\mathtt{n}}{\mathtt{p}\left(\sigma \right)}}{\mathsf{\vartheta }}_{\mathsf{\eta }}\left(\mathsf{\mu }\right){\mathsf{\chi }}_{\mathtt{B}(\sigma ,\mathtt{r})}\right|}{2{\mathsf{\eta }}_2{\kappa }_{\mathsf{\eta }}^{\frac{1}{\mathtt{q}\left(\mathsf{\mu }\right)}}}}\right)}^{\mathtt{p}\left(\mathsf{\mu }\right)}\mathtt{d}\mathsf{\mu }\le 1.\hfill \end{array}$$

(53)

Consequently, in both scenarios, the proof of (43) is finished. It is now necessary to investigate the intermediate term and show that it is either smaller than or equal to the inequality’s final component.

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}{∥(1-\mathsf{\sigma }){\xi }_{\mathsf{\eta }}+\mathsf{\sigma }{\mathsf{\vartheta }}_{\mathsf{\eta }}∥}_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}(·),\mathtt{v}(·)}\right)}\mathtt{d}\mathsf{\sigma }\hfill \\ \hfill & \le {\int }_{\mathsf{\Omega }}(1-\mathsf{\sigma }){∥{\xi }_{\mathsf{\eta }}∥}_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}(·),\mathtt{v}(·)}\right)}\mathtt{d}\mathsf{\sigma }+\mathsf{\sigma }{∥{\mathsf{\vartheta }}_{\mathsf{\eta }}∥}_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}(·),\mathtt{v}(·)}\right)}\mathtt{d}\mathsf{\sigma }\hfill \\ \hfill & \le {\displaystyle \frac{{∥{\xi }_{\mathsf{\eta }}∥}_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}(·),\mathtt{v}(·)}\right)}}{2}}+{\displaystyle \frac{{∥{\mathsf{\vartheta }}_{\mathsf{\eta }}∥}_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}(·),\mathtt{v}(·)}\right)}}{2}}={\displaystyle \frac{\parallel {\xi }_{\mathsf{\eta }}{\parallel }_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}(·),\mathtt{v}(·)}\right)}+{\parallel {\mathsf{\vartheta }}_{\mathsf{\eta }}\parallel }_{{\ell }_{\mathtt{q}(·)}\left({\mathtt{M}}_{\mathtt{p}(·),\mathtt{v}(·)}\right)}}{2}}.\hfill \end{array}$$

(54)

Taking into account Equations (53) and (54), we obtain required finding (43). □

Remark 2. 

If we take $\mathtt{v}\left(\sigma \right)=\mathtt{p}\left(\sigma \right),\mathtt{q}(.)=\infty$, and ${\mathtt{p}}^{+}={\mathtt{p}}^{-}=\mathtt{p}$, then it refines the double inequality in classical Lebesgue space (see [85], p. 51).

Our result also provides refinement to the Hermite–Hadamard inequality presented in [83] for classical ${\mathtt{L}}_{\mathtt{p}}$ spaces:

$${∥{\displaystyle \frac{{\xi }_{\mathsf{\eta }}+{\mathsf{\vartheta }}_{\mathsf{\eta }}}{2}}∥}_{{\mathtt{L}}_{\mathtt{p}}(·)}\le {\int }_{\mathsf{\Omega }}{∥(1-\mathsf{\sigma }){\xi }_{\mathsf{\eta }}+\mathsf{\sigma }{\mathsf{\vartheta }}_{\mathsf{\eta }}∥}_{{\mathtt{L}}_{\mathtt{p}}(·)}\mathtt{d}\mathsf{\sigma }\le {\displaystyle \frac{\parallel {\xi }_{\mathsf{\eta }}{\parallel }_{{\mathtt{L}}_{\mathtt{p}}(·)}+{\parallel {\mathsf{\vartheta }}_{\mathsf{\eta }}\parallel }_{{\mathtt{L}}_{\mathtt{p}}(·)}}{2}}.$$

If we take $\mathtt{p}(.)=\mathtt{p}$ and $\mathtt{q}(.)=\mathtt{q}$, we have the double inequality in classical ${\ell }_{\mathtt{q}}\left({\mathtt{M}}_{\mathtt{p},\mathtt{v}}\right)$ space, which is also new.

8. Conclusions and Future Directions

Tensorial Hilbert spaces and their inequalities are key topics in functional analysis, quantum mechanics, and other fields of mathematics and physics. In this note, we construct various new identities and lemmas by using the Stone–Weierstrass theorem, as well as Newton–Milne inequalities and convexity involving the arithmetic–geometric mean type. Inequalities are developed by utilizing the spectral resolution of self-adjoint operators in Hilbert spaces. Additionally, we determine upper bounds for these inequalities and give additional examples and consequences for transcendental functions using different kinds of generalized convex mappings.

Furthermore, we used mixed-norm Moore spaces with variable exponent functions to develop a Hermite–Hadamard inequality via a novel and important approach that has never been performed before with any other kind of function space. As we apply new restrictions on exponent functions using different forms of exponent functions, we show that it refines and generalizes numerous other relevant inequalities. Such types of mathematical inequalities supporting tensor Hilbert and Moore spaces are rarely examined in the literature, so we believe that this work opens up a whole new avenue in mathematical inequality theory.