Tensorial Maclaurin Approximation Bounds and Structural Properties for Mixed-Norm Orlicz–Zygmund Spaces

Abstract

This article explores two distinct function spaces: Hilbert spaces and mixed-Orlicz–Zygmund spaces with variable exponents. We first examine the relational properties of Hilbert spaces in a tensorial framework, utilizing self-adjoint operators to derive key results. Additionally, we extend a Maclaurin-type inequality to the tensorial setting using generalized convex mappings and establish various upper bounds. A non-trivial example involving exponential functions is also presented. Next, we introduce a new function space, the mixed-Orlicz–Zygmund space $\left({\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)\right)$, which unifies Orlicz–Zygmund spaces of integrability and sequence spaces. We investigate its fundamental properties including separability, compactness, and completeness, demonstrating its significance. This space generalizes the existing structures, reducing to mixed-norm Lebesgue spaces when $\beta =0$ and to classical Lebesgue spaces when $\mathtt{q}=\infty ,\beta =0$. Given the limited research on such spaces, our findings contribute valuable insights to the functional analysis.

1. Introduction

Inequalities in function spaces are a cornerstone of functional analysis and the associated disciplines, which give basic constraints relating the functions, integrals, derivatives, and their norms. They are used extensively to study the behavior of functions in a wide variety of mathematical fields. For example, Sobolev and Poincaré inequalities have been used to prove the existence and uniqueness of PDE solutions [1], while Hölder’s and Minkowski’s inequalities are used to prove operator boundedness [2]. In addition, inequalities are a key tool in energy functional minimization and solving variational problems [3], approximating differential operators’ eigenvalues via the Rayleigh quotient [4], and proving the stability of numerical schemes using norm inequalities [5].

The literature encompasses, in depth, a number of integral inequalities, e.g., Hermite–Hadamard, Giaccardi, Jensen, midpoint, Young’s, Trapezoid, Simpson, Ostrowski, and so forth. For example, the Hermite–Hadamard inequality has been studied in many directions, e.g., its stochastic counterpart [6], fuzzy-valued mappings [7], center–radius order relations [8], inclusion order [9], and Hilbert spaces [10]. Similarly, other inequalities have been explored using similar techniques, e.g., the Ostrowski-type inequality, which has been generalized in stochastic environments [11] under various order relations [12,13] and in function spaces [14]. For further information on different inequalities, we refer to [15,16,17,18,19] and the references therein.

The Maclaurin series with an integral remainder term is an effective tool for approximating functions and expressing the approximation error precisely. The integral remainder form is particularly useful in theoretical and numerical approximations when error control is crucial.

1.1. Benefits and Significance of the Maclaurin Series Approximation

• Function Approximation: Replace complex functions with simpler polynomial expressions, which are utilized in calculators and computer software to evaluate functions with high precision.
• Economics and Finance: Investigate physical systems near reference states and use Hamiltonians to solve quantum systems with perturbative relativistic formulas for low velocities.
• Numerical Methods: Use expansions to generate numerical differentiation formulas and analyze truncation mistakes to ensure accuracy.
• Machine Learning: Optimize loss functions and enhance algorithms. Gradient descent inequality employs first- and second-order approximations to assess convergence.
• Physics and Engineering: Approximate potential energy functions at equilibrium points are used to expand state equations or partition functions in systems close to equilibrium.

For more applications in various disciplines, we refer to the following articles [20,21,22]. This widely recognized equality can be stated as follows.

Theorem 1. 

Let $\zeta :\mathsf{R}\to \mathsf{R}$ be a function on some set $\Omega \subseteq \mathsf{R}$ with $\mathtt{n}+\mathtt{1}$ continuous derivatives. Then, for any $\nu \in \Omega$, we have

$$\begin{array}{c}\hfill \zeta (\nu )=\zeta \left(0\right)+{\zeta }^{\prime }\left(0\right)\nu +{\displaystyle \frac{{\zeta }^{″}\left(0\right)}{2!}}{\nu }^2+\dots +{\displaystyle \frac{{\zeta }^{\left(\mathtt{n}\right)}\left(0\right)}{\mathtt{n}!}}{\nu }^{\mathtt{n}}+{\mathtt{R}}_{\mathtt{n}}(\nu ),\end{array}$$

(1)

where ${\mathtt{R}}_{\mathtt{n}}(\nu )$ is the remainder term after $\mathtt{n}$ terms. The remainder term ${\mathtt{R}}_{\mathtt{n}}(\nu )$ can be expressed in integral form as

$${\mathtt{R}}_{\mathtt{n}}(\nu )={\displaystyle \frac{1}{\mathtt{n}!}}{\int }_0^{\nu }{(\nu -\mathtt{t})}^{\mathtt{n}}{\zeta }^{(\mathtt{n}+\mathtt{1})}\left(\mathtt{t}\right)\mathtt{dt}.$$

This formula shows that the error ${\mathtt{R}}_{\mathtt{n}}(\nu )$ depends on the $(\mathtt{n}+\mathtt{1})$-th derivative of $\zeta (\nu )$ and the behavior of $\zeta (\nu )$ on the interval $[0,\nu ]$.

Variable exponent spaces and Hilbert spaces are inherent mathematical structures of varied use across various disciplines. Their joint application strengthens the analysis of complex systems via a more efficient and effective approach. They are, for example, essential for modeling relations in complex systems where evidence is characterized by extreme variability [23]. They enhance approximation in approximation theory through the improved representation of functions via integral operators [24].

Variable exponent spaces are used in partial differential equations (PDEs) with non-traditional growth conditions [25] and the study of nonlinear equations, especially in dynamical systems, where normal techniques break down due to irregularities or complexity [26]. Hilbert spaces, nonetheless, are the bedrock of quantum mechanics, wherein quantum states and observables are constructed within their theory [27]. Hilbert spaces are also extensively employed in signal processing and wireless communications, where linear operator algebra must be used in signal analysis [28]. Moreover, statistical inference has a lot to do with Gaussian Hilbert space measures [29], through which statisticians can place bounds on approximations and the optimization of linear rules under various constraints. For recent developments in other areas, we refer to [30,31].

The significance of variable exponent spaces and Hilbert spaces has led to numerous original results and well-known inequalities in such spaces. Hilbert spaces and tensor operations have been utilized by Frank Hansen and Huzihiro Araki [32] in proving Jensen’s operator-type inequalities. In [33], the authors compared the two spaces side by side and derived numerous striking results related to Hadamard and triangular-type inequalities by utilizing various unified and generalized convex mappings. Prasenjit and Tapas Kumar [34] also worked with frames in the tensor product space of n-Hilbert spaces and generalized earlier results from bases to frames and dual frames in tensor product spaces.

Theorem 2 

([34]). Assume ${\left\{{\mathtt{m}}_{\mathtt{i}}\right\}}_{\mathtt{i}=\mathtt{1}}^{\infty }$ and ${\left\{{\mathtt{n}}_{\mathtt{j}}\right\}}_{\mathtt{j}=\mathtt{1}}^{\infty }$ be the class of vectors contained in ${\mathtt{H}}_{\mathtt{1}}$ and ${\mathtt{H}}_{\mathtt{2}}$ Hilbert spaces. The vector product ${\left\{{\mathtt{m}}_{\mathtt{i}}\otimes {\mathtt{n}}_{\mathtt{j}}\right\}}_{\mathtt{i},\mathtt{j}=\mathtt{1}}^{\infty }\subseteq {\mathtt{H}}_{\mathtt{1}}\otimes {\mathtt{H}}_{\mathtt{2}}$ is a frame related to $\left({\sigma }_2\otimes {\kappa }_2,\dots ,{\sigma }_{\mathtt{n}}\otimes {\kappa }_{\mathtt{n}}\right)$ for ${\mathtt{H}}_{\mathtt{1}}\otimes {\mathtt{H}}_{\mathtt{2}}$ iff ${\left\{{\mathtt{m}}_{\mathtt{i}}\right\}}_{\mathtt{i}=\mathtt{1}}^{\infty }$ is a frame related to $\left({\sigma }_2,\dots ,{\sigma }_{\mathtt{n}}\right)$ for ${\mathtt{H}}_{\mathtt{1}}$ and ${\left\{{\mathtt{n}}_{\mathtt{j}}\right\}}_{\mathtt{j}=\mathtt{1}}^{\infty }$ is a frame related to $\left({\kappa }_2,\dots ,{\kappa }_{\mathtt{n}}\right)$ for ${\mathtt{H}}_{\mathtt{2}}$. Since ${\left\{e_i\right\}}_{\mathtt{i}=\mathtt{1}}^{\infty }$ and ${\left\{{\mathtt{k}}_{\mathtt{j}}\right\}}_{\mathtt{j}=\mathtt{1}}^{\infty }$ are dual frames corresponding to $\left({\sigma }_2,\dots ,{\sigma }_{\mathtt{n}}\right)$ and $\left({\kappa }_2,\dots ,{\kappa }_{\mathtt{n}}\right)$ of ${\left\{{\mathtt{m}}_{\mathtt{i}}\right\}}_{\mathtt{i}=\mathtt{1}}^{\infty }$ and ${\left\{{\mathtt{n}}_{\mathtt{j}}\right\}}_{\mathtt{j}=\mathtt{1}}^{\infty }$, respectively, for all $p\in {\mathtt{H}}_{\mathtt{1}},q\in {\mathtt{H}}_{\mathtt{2}}$,

$$\mathtt{f}=\sum _{\mathtt{i}=\mathtt{1}}^{\infty }{⟨\mathtt{f},{\mathtt{e}}_{\mathtt{i}}\mid {\sigma }_2,\dots ,{\sigma }_{\mathtt{n}}⟩}_1{\mathtt{m}}_{\mathtt{i}},\mathrm{and}\mathtt{g}=\sum _{\mathtt{j}=\mathtt{1}}^{\infty }{⟨\mathtt{g},{\mathtt{k}}_{\mathtt{j}}\mid {\kappa }_2,\dots ,{\kappa }_{\mathtt{n}}⟩}_2{\mathtt{n}}_{\mathtt{j}}.$$

Then, for all $\mathtt{f}\otimes \mathtt{g}\in {\mathtt{H}}_{\mathtt{1}}\otimes {\mathtt{H}}_{\mathtt{2}}$, we have

$$\begin{array}{cc}& \mathtt{f}\otimes \mathtt{g}\hfill \\ & =\left(\sum _{\mathtt{i}=\mathtt{1}}^{\infty }{⟨\mathtt{f},{\mathtt{e}}_{\mathtt{i}}\mid {\sigma }_2,\dots ,{\sigma }_{\mathtt{n}}⟩}_1{\mathtt{m}}_{\mathtt{i}}\right)\otimes \left(\sum _{\mathtt{j}=\mathtt{1}}^{\infty }{⟨\mathtt{g},{\mathtt{k}}_{\mathtt{j}}\mid {\kappa }_2,\dots ,{\kappa }_{\mathtt{n}}⟩}_2{\mathtt{n}}_{\mathtt{j}}\right)\hfill \\ & =\sum _{\mathtt{i},\mathtt{j}=\mathtt{1}}^{\infty }{⟨\mathtt{f},{\mathtt{e}}_{\mathtt{i}}\mid {\sigma }_2,\dots ,{\sigma }_{\mathtt{n}}⟩}_1{⟨\mathtt{g},{\mathtt{k}}_{\mathtt{j}}\mid {\kappa }_2,\dots ,{\kappa }_{\mathtt{n}}⟩}_2\left({\mathtt{m}}_{\mathtt{i}}\otimes {\mathtt{n}}_{\mathtt{j}}\right).\hfill \end{array}$$

Vuk Stojiljkovic [35,36] used classical integral operators and twice-differentiable mappings to the continuous functions on self-adjoint operators in Hilbert space to develop Simpson and Ostrowski-type double inequalities.

Theorem 3 

([36]). Let $\mathcal{U}$ and $\mathcal{V}$ be self-adjoint operators, and let $\mathcal{SP}\left(\mathcal{U}\right),\mathcal{SP}\left(\mathcal{V}\right)\subset \Delta$ be their associated spectrums. Assuming that $\zeta$ is a continuous function on $\Delta ,$ we obtain

$$\begin{array}{cc}\hfill & {\int }_0^1\zeta ((1-\alpha )\mathcal{U}\otimes 1+\alpha 1\otimes \mathcal{V})\mathtt{d}\alpha -\zeta \left({\displaystyle \frac{\mathcal{U}\otimes 1+1\otimes \mathcal{V}}{2}}\right)\hfill \\ \hfill =& {\displaystyle \frac{{(1\otimes \mathcal{V}-\mathcal{U}\otimes 1)}^2}{16}}\left[{\int }_0^1{\alpha }^2{\zeta }^{″}\left(\left(1-\alpha \right)\mathcal{U}\otimes 1+\alpha 1\otimes \mathcal{V}\right)\mathtt{d}\alpha \right.\hfill \\ \hfill & \left.+{\int }_0^1{(\alpha -1)}^2{\zeta }^{″}\left(\left({\displaystyle \frac{1-\alpha }{2}}\right)\mathcal{U}\otimes 1+\left({\displaystyle \frac{1+\alpha }{2}}\right)1\otimes \mathcal{V}\right)\mathtt{d}\alpha \right].\hfill \end{array}$$

Dragomir [37] investigated multiple bounds and developed a well-known classical trapezoid-type inequality in the context of tensor Hilbert spaces.

Theorem 4 

([37]). Let $\mathcal{U}$ and $\mathcal{V}$ be self-adjoint operators, and let $\mathcal{SP}\left(\mathcal{U}\right),\mathcal{SP}\left(\mathcal{V}\right)\subset \Delta$ be their associated spectrums. Assuming that $\zeta$ is a continuous function on $\Delta ,$ we obtain

$$\begin{array}{cc}& (1-\alpha )1\otimes \zeta \left(\mathcal{V}\right)+\alpha \zeta \left(\mathcal{U}\right)\otimes 1-{\int }_0^1\zeta ((1-\mathtt{u})\mathcal{U}\otimes 1+\mathtt{u}1\otimes \mathcal{V})\mathtt{d}\mathtt{u}\hfill \\ & =(1\otimes \mathcal{V}-\mathcal{U}\otimes 1){\int }_0^1(\mathtt{u}-\alpha ){\zeta }^{\prime }((1-\mathtt{u})\mathcal{U}\otimes 1+\mathtt{u}1\otimes \mathcal{V})\mathtt{d}\mathtt{u}.\hfill \end{array}$$

for all $\alpha \in [0,1]$.

Khan et al. [38] used fractional integral operators and several forms of extended convex mappings to find boundaries for Simpson and Hermite–Hadamard inequalities.

Theorem 5 

([38]). Let $\mathcal{U}$ and $\mathcal{V}$ be self-adjoint operators, and let $\mathcal{SP}\left(\mathcal{U}\right),\mathcal{SP}\left(\mathcal{V}\right)\subset \Delta$ be their associated spectrums. Assuming that $\zeta$ is a continuous function on $\Delta ,$ we obtain

$$\begin{array}{cc}& \parallel {\displaystyle \frac{\mathcal{V}\otimes 1+1\otimes \mathcal{U}}{2}}-{\displaystyle \frac{2(\mathcal{U}\otimes 1-\mathcal{V}\otimes 1)}{{(\sqrt{\mathcal{U}\otimes 1}-\sqrt{\mathcal{V}\otimes 1})}^2}}{\int }_0^1\zeta ((1-\mathtt{u})\mathcal{V}\otimes 1+\mathtt{u}\mathcal{U}\otimes 1))\mathtt{d}\mathtt{u}\parallel \hfill \\ & \le {\displaystyle \frac{{(\sqrt{1\otimes \mathcal{U}}-\sqrt{\mathcal{V}\otimes 1})}^2}{4}}{\displaystyle \frac{\left(∥{\zeta }^{\prime }\left(\mathcal{U}\right)∥+∥{\zeta }^{\prime }\left(\mathcal{V}\right)∥\right)}{2}}.\hfill \end{array}$$

(2)

For more current results related to the results in this article, see [38,39,40,41,42,43,44,45,46] and their references.

Variable exponent Lebesgue spaces $\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$ are a modern mathematical analysis field that developed from the conventional Lebesgue space. In 1931, W. Orlicz [47] developed sequence spaces ${\ell }^{{\mathtt{p}}_{\mathtt{n}}}$ with variable exponent $\left({\mathtt{p}}_{\mathtt{n}}\right)$ and discussed some of their properties. In the same paper, Orlicz defined the class of measurable functions $\zeta$ such that ${\int }_0^1{|\zeta \left(\mathtt{x}\right)|}^{\mathtt{p}(\mu )}\mathtt{dx}<\infty$, thereby generalizing the concept of classical Lebesgue spaces of integrability. In this work, they let the exponent $\mathtt{p}(\mu )$ depend on the point $\mathtt{x}$ in the domain ∆. Their adaptability makes them better suitable for simulating physical processes, in which the space’s behavior may vary locally in a nonlinear way. Though ${\mathtt{L}}^{\mathtt{p}(·)}$ first emerged in 1931, they were initially analyzed as Banach spaces in [48].

Taking inspiration from these various authors, they constructed and generalized numerous other forms of function spaces and examined their features and boundedness of operators with varied exponents. For instance, in [49], the authors used spaces of variable integrability to study uniform convexity; in [50], they studied some interesting problems related to the fixed point theorem in the varying exponent spaces; in [51], they examined the boundedness of certain operators on grand Herz spaces with variable exponents; in [52], they examined interpolation theorems for variable exponent Lebesgue spaces; and in [53], they examined modular-proximal gradient algorithms in varying exponent spaces. Fernanda and James [54] investigated the algebraic and topological reflexivity features of sequence spaces with varying exponents. Amri et al. [55] employed modular function spaces to examine fixed point theorem with application in the space of variable integrability. Haim Brezis [56] explored different kinds of function spaces and discussed problems related to reflexive spaces, separable spaces, and uniform convexity. Zhenghua Luo and Qingjin Cheng [57] investigated a novel convexity property of convex functions, resulting in a characterization of the class of reflexive Banach spaces. Jain et al. [58] investigated quasi-reflexivity and the sup of linear functions with interesting applications. Enrico Pasqualetto and Tapio Rajala [59] studied first-order Sobolev spaces on reflexive Banach spaces, using relaxation, test plans, and divergence.

Completeness, compactness, and the separability of variable exponent spaces play a very essential role in several areas, such as checking the boundedness of operators, checking the existence of the solution of a partial differential equation, embedding of spaces, etc. For instance, the authors examined these three properties for Herz spaces in [60], demonstrated separation axioms, representation theorems, and compactness in [61], examined completeness for Moore-closed spaces and centered bases in [62], determined compactness and separability in [63], and demonstrated several pertinent results in weighted Lorentz spaces using the Hardy Operator in [64].

Bandaliyev et al. [65] studied the precompactness and completeness of sets in variable exponent Morrey spaces using bounded metric measure spaces.

Theorem 6 

([65]). Let $(\Omega ,\rho ,\nu )$ be a bounded doubling metric measure space, and let $\mathtt{p}(·)\in \mathcal{P}(\Omega ,\nu )$ satisfy $0<{\mathtt{p}}_{-}\le {\mathtt{p}}_{+}<\infty$ and ${sup}_{\sigma \in \Omega }{\displaystyle \frac{\alpha (\sigma )}{\mathtt{p}(\sigma )}}<{\displaystyle \frac{1}{{\mathtt{p}}_{+}}}$. Assume that a subset $\mathcal{Q}$ of ${\mathtt{L}}^{\mathtt{p}(·),\alpha (·)}(\Omega )$ satisfies the following:

• $\mathcal{Q}$ is bounded in ${\mathtt{L}}^{\mathtt{p}(·),\alpha (·)}(\Omega )$;
• For some $\mathtt{q}\in (0,{\mathtt{p}}_{-})$,

$$\underset{\tau \to 0^{+}}{lim}\underset{\zeta \in \mathcal{Q}}{sup}{∥{\int }_{\mathcal{B}(·,\tau )}{|\zeta (·)-\zeta \left(\mathtt{y}\right)|}^{\mathtt{q}}\mathrm{d}\nu \left(\mathtt{y}\right)∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}}},\alpha (·)}(\Omega )}=0.$$

Then, $\mathcal{Q}$ is totally bounded in ${\mathtt{L}}^{\mathtt{p}(·),\alpha (·)}(\Omega )$.

Orlicz–Zygmund spaces are a class of function spaces that generalize both Orlicz spaces and Zygmund spaces. They enable logarithmic changes to the evolution of Orlicz functions, making them helpful in a variety of applications including nonlinear analysis and partial differential equations. For instance in [66], the authors investigated the Lipschitz regularity for local minimizers of convex variational integrals of the form using Orlicz–Zygmund spaces with variable exponents; in [67], the authors investigated the higher differentiability of minimizers of variational integrals with variable Orlicz–Zygmund spaces; in [68], the authors investigated global gradient estimates for the borderline case of double-phase problems with BMO for Orlicz–Zygmund spaces; and in [69], the authors investigated regularity for double-phase energy functionals of different power growth.

For further recent results relevant to variable exponent spaces, see [70,71,72,73,74,75,76] and the references therein.

1.2. Main Contribution

The article makes two significant contributions. The first is that tensor approximation can be used to extend equality (1) from the lower dimension (classical sense) to a higher dimension since tensor approximation deals with a wide range of complex functions in a very straightforward manner. Second, we can introduce a new norm structure, as Orlicz–Zygmund spaces with variable exponents have been extensively studied for analyzing problems involving variable growth conditions and non-standard behavior in partial differential equations.

The Zygmund space ${\mathtt{L}}^{\mathtt{p}}{log}^{\beta }\mathtt{L}\left(\Omega ;{\mathsf{R}}^{\mathtt{n}}\right)$, for $1⩽\mathtt{p}<\infty ,\beta \in \mathsf{R}(\beta ⩾0$ for $\mathtt{p}=1)$ is defined as the Orlicz–Zygmund space ${\mathtt{L}}^{\Phi }\left(\Omega ;{\mathsf{R}}^{\mathsf{n}}\right)$ generated by the Young function

$$\Phi (\mu )\simeq {\mu }^{\mathtt{p}}{log}^{\beta }(\mathtt{e}+\mu )\mathrm{for}\mathrm{every}\mu \ge {\mu }_0\ge 0.$$

Therefore, the Orlicz–Zygmund space ${\mathtt{L}}^{\mathtt{p}}{log}^{\beta }\mathtt{L}\left(\Omega ;{\mathsf{R}}^{\mathtt{n}}\right)$ is defined as

$${\mathtt{L}}^{\mathtt{p}}{log}^{\beta }\mathtt{L}\left(\Omega ;{\mathsf{R}}^{\mathtt{n}}\right):=\left\{\zeta :\Omega \to {\mathsf{R}}^{\mathtt{n}}\mathrm{measurable}:{\int }_{\Omega }{|\zeta |}^{\mathtt{p}}{log}^{\beta }(\mathtt{e}+|\zeta \left|\right)\mathtt{d}\nu <\infty \right\}$$

and it becomes a Banach space with the Luxemburg norm

$${\parallel \zeta \parallel }_{\mathtt{L}{log}^{\beta }\mathtt{L}\left(\Omega ;{\mathsf{R}}^{\mathtt{n}}\right)}:=inf\left\{\alpha >0:{\zeta }_{\Omega }{\left|{\displaystyle \frac{\zeta }{\alpha }}\right|}^{\mathtt{p}}{log}^{\beta }\left(\mathtt{e}+\left|{\displaystyle \frac{\zeta }{\alpha }}\right|\right)\mathtt{d}\nu \le 1\right\}.$$

The main idea we use is a combination of the variable sequence space $\left({\ell }_{\mathtt{q}(·)}\right)$ and variable Orlicz–Zygmund space $\left({\mathtt{L}}^{\mathtt{p}}{log}^{\beta }\mathtt{L}\left(\Omega ;{\mathsf{R}}^{\mathtt{n}}\right)\right)$ that is merged simultaneously, and we introduce a new type of modular functional with an associated norm that is based on these two spaces. In this space, we define a new norm structure that we call mixed-variable Orlicz–Zygmund space $\left({\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)\right)$ and investigate several well-known properties, such as completeness, separability, and precompactness. Since these types of spaces related to developed results are rare in the literature, this paper will play a very important role in the advancement of results related to these in function spaces.

As a main motivation for this study, we were primarily influenced by the works of the following authors [32,35,65,66].

The work is organized into six sections. In Section 2, we will recall some basic definitions and arithmetic operations in tensor Hilbert spaces. In Section 3, we construct our major conclusions, including Maclaurin-type approximation, using various types of generalized convex mappings in tensorial Hilbert spaces. In Section 4, we define a few well-known notions and results that we need to support our primary conclusions regarding Orlicz–Zygmund spaces and other related spaces. In Section 5, we demonstrate the separability, completeness, and compactness of newly defined mixed-Orlicz–Zygmund spaces. In Section 6, we discuss our main findings and some future possible work related to these results.

2. Preliminaries

In this section, we recall some fundamental concepts related to extended convex mappings and arithmetic operations in tensor Hilbert spaces. The following articles [32,39] contain further important concepts and discoveries related to this section.

Definition 1 

([33]). An inner product on a complex linear space $\mathcal{K}$ is a map

$$(·,·):\mathcal{K}\times \mathcal{K}\to \mathtt{C}$$

such that for all ${\xi }_1,{\xi }_2,{\xi }_3\in \mathcal{K}$ and $\alpha \in \mathtt{C}$, we have

$$\begin{array}{cc}\hfill ⟨{\xi }_1+{\xi }_2,{\xi }_3⟩& =⟨{\xi }_1,{\xi }_3⟩+⟨{\xi }_2+{\xi }_3⟩\hfill \\ \hfill ⟨\alpha {\xi }_1,{\xi }_2⟩& =\alpha ⟨{\xi }_1,{\xi }_2⟩\hfill \\ \hfill ⟨{\xi }_1,{\xi }_2⟩& =\overline{⟨{\xi }_2,{\xi }_1⟩}\hfill \\ \hfill ⟨{\xi }_1,{\xi }_1⟩& \ge 0,⟨{\xi }_1,{\xi }_1⟩=0⟺{\xi }_1=0.\hfill \end{array}$$

A complete inner product space is called a Hilbert space.

Definition 2 

([33]). Let $\zeta :\mathcal{U}\times \mathcal{V}\to \mathcal{K}$ be a bilinear mapping. The related tensor product of $\mathcal{U}$ with $\mathcal{V}$ on Hilbert space $\mathcal{K}$ is defined as follows:

• The Hilbert space $\mathcal{K}$ contains the class of $\zeta ({\xi }_1,{\xi }_2)({\xi }_1\in \mathcal{U},{\xi }_2\in \mathcal{V})$ all vectors, and $\mathcal{K}$ is its span;
• $\left(\zeta \left({\xi }_1,{\xi }_2\right)\mid \zeta \left({\xi }_3,{\xi }_4\right)\right)=\left({\xi }_1\mid {\xi }_2\right)\left({\xi }_3\mid {\xi }_4\right)$ for ${\xi }_1,{\xi }_2\in \mathcal{U},{\xi }_3,{\xi }_4\in \mathcal{V}$. If $(\mathcal{K},\zeta )$ is a Cartesian product of $\mathcal{U}$ and $\mathcal{V}$, it is typical to read as ${\xi }_1\otimes {\xi }_2$ instead of $\zeta ({\xi }_1,{\xi }_2)$. A Cartesian product $\mathcal{U}\otimes \mathcal{V}$ and a function $({\xi }_1,{\xi }_2)\mapsto {\xi }_1\otimes {\xi }_2$ of $\mathcal{U}\times \mathcal{V}$ into $\mathcal{U}\otimes \mathcal{V},$ satisfy the following axioms

  $$\begin{array}{cc}\hfill \left({\xi }_1+{\xi }_2\right)\otimes {\xi }_2& ={\xi }_1\otimes {\xi }_2+{\xi }_2\otimes {\xi }_2\hfill \\ \hfill (\alpha {\xi }_1)\otimes {\xi }_2& =\alpha ({\xi }_1\otimes {\xi }_2)\hfill \\ \hfill {\xi }_1\otimes \left({\xi }_3+{\xi }_4\right)& ={\xi }_1\otimes {\xi }_3+{\xi }_1\otimes {\xi }_3\hfill \\ \hfill {\xi }_1\otimes (\alpha {\xi }_2)& =\alpha ({\xi }_1\otimes {\xi }_2),\hfill \end{array}$$

  where $\alpha \in \mathcal{K}.$

Let $\zeta :{\Delta }_1\times \dots \times {\Delta }_{\mathtt{p}}\to \mathsf{R}$ is taken to be bounded mapping defined in terms of intervals. Assume that $\wp =\left({\wp }_1,\dots ,{\wp }_{\mathtt{p}}\right)$ be a $\mathtt{p}$-tuple of the self-adjoint operators associated with ${\mathtt{E}}_1,\dots ,{\mathtt{E}}_{\mathtt{p}}$ Hilbert spaces. Then,

$${\wp }_{\mathtt{i}}={\int }_{{\Delta }_{\mathtt{i}}}{\alpha }_{\mathtt{i}}{\mathtt{dE}}_{\mathtt{i}}\left({\alpha }_{\mathtt{i}}\right)$$

is the spectrum of possible operators for $\mathtt{i}=1,\dots ,\mathtt{p}$; following [32], we define ${\wp }_{\mathtt{i}}$ as follows:

$$\zeta \left({\wp }_1,\dots ,{\wp }_{\mathtt{p}}\right):={\int }_{{\Delta }_1}\dots {\int }_{{\Delta }_{\mathtt{p}}}\zeta \left({\alpha }_1,\dots ,{\alpha }_{\mathtt{p}}\right){\mathtt{dE}}_1\left({\alpha }_1\right)\otimes \dots \otimes {\mathtt{dE}}_{\mathtt{p}}\left({\alpha }_{\mathtt{p}}\right).$$

When the dimensions of Hilbert spaces are finite, many complicated processes can be considerably reduced by integrating processes into finite summations. The authors [77] expand the construction, and it is defined as follows:

$$\zeta \left({\wp }_1,\dots ,{\wp }_{\mathtt{p}}\right)={\zeta }_1\left({\wp }_1\right)\otimes \dots \otimes {\zeta }_{\mathtt{p}}\left({\wp }_{\mathtt{p}}\right),$$

where $\zeta$ can be split as a product of one-variable mappings $\zeta \left({\mathtt{a}}_1,\dots ,{\mathtt{a}}_{\mathtt{p}}\right)={\zeta }_1\left({\mathtt{a}}_1\right)\dots {\zeta }_{\mathtt{p}}\left({\mathtt{a}}_{\mathtt{p}}\right)$.

If $\zeta$ is sub(super)-multiplicative across the interval $\Delta$, then

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

and additionally, if $\zeta$ is continuous on $[0,\infty )$, one has

$$\zeta (\mathcal{U}\otimes \mathcal{V})\ge (\le )\zeta \left(\mathcal{U}\right)\otimes \zeta \left(\mathcal{V}\right)\mathrm{for}\mathrm{all}\mathcal{U},\mathcal{V}\ge 0.$$

Consequently, it may be shown that if

$$\mathcal{U}={\int }_{[0,\infty )}{\xi }_1\mathtt{dE}\left({\xi }_1\right)\mathrm{and}\mathcal{V}={\int }_{[0,\infty )}{\xi }_2\mathtt{dF}\left({\xi }_2\right),$$

where $\mathcal{U}$ and $\mathcal{V}$ are spectral resolutions for function $\zeta$ on $[0,\infty )$, then

$$\zeta (\mathcal{U}\otimes \mathcal{V})={\int }_{[0,\infty )}{\int }_{[0,\infty )}\zeta \left({\xi }_1{\xi }_2\right)\mathtt{dE}\left({\xi }_1\right)\otimes \mathtt{dF}\left({\xi }_2\right).$$

The geometric property of linear bounded operator $\mathcal{U},\mathcal{V}>0$ is represented as follows:

$$\mathcal{U}{\#}_{\mathtt{p}}\mathcal{V}:={\mathcal{U}}^{1/2}{\left({\mathcal{U}}^{-1/2}\mathcal{V}{\mathcal{U}}^{-1/2}\right)}^{\mathtt{p}}{\mathcal{U}}^{1/2},$$

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

$$\mathcal{U}\#\mathcal{V}:={\mathcal{U}}^{1/2}{\left({\mathcal{U}}^{-1/2}\mathcal{V}{\mathcal{U}}^{-1/2}\right)}^{1/2}{\mathcal{U}}^{1/2}.$$

By the definitions of # and ⊗, we have

$$\mathcal{U}\#\mathcal{V}=\mathcal{V}\#\mathcal{U}\mathrm{and}(\mathcal{U}\#\mathcal{V})\otimes (\mathcal{V}\#\mathcal{U})=(\mathcal{U}\otimes \mathcal{V})\#(\mathcal{V}\otimes \mathcal{U}).$$

Examine the final result that is comparable to the tensorial product that is

$$\left(\mathcal{U}\beta \right)\otimes \left(\mathcal{V}\mathcal{S}\right)=(\mathcal{U}\otimes \mathcal{V})(\mathcal{U}\otimes \mathcal{S}),$$

that holds $\forall \mathcal{U},\mathcal{V},\beta ,\mathcal{S}\in \mathtt{B}\left(\mathtt{H}\right)$. If we take $\beta =\mathcal{U}$ and $\mathcal{S}=\mathcal{V}$, then we obtain

$${\mathcal{U}}^2\otimes {\mathcal{V}}^2={(\mathcal{U}\otimes \mathcal{V})}^2.$$

Through induction, we have

$${\mathcal{U}}^{\sigma }\otimes {\mathcal{V}}^{\sigma }={(\mathcal{U}\otimes \mathcal{V})}^{\sigma }\mathrm{for}\mathrm{any}\mathrm{natural}\mathrm{number}\sigma \ge 0.$$

Specifically,

$${\mathcal{U}}^{\sigma }\otimes 1={(\mathcal{U}\otimes 1)}^{\sigma }\mathrm{and}1\otimes {\mathcal{V}}^{\sigma }={(1\otimes \mathcal{V})}^{\sigma },$$

for all $\sigma \ge 0$. Additionally, we note that the $1\otimes \mathcal{V}$ and $\mathcal{U}\otimes 1$ are commutative with each other that is

$$(1\otimes \mathcal{V})(\mathcal{U}\otimes 1)=\mathcal{U}\otimes \mathcal{V}=(\mathcal{U}\otimes 1)(1\otimes \mathcal{V}).$$

Moreover, for any two natural numbers ${\sigma }_1,{\sigma }_2$

$${(\mathcal{U}\otimes 1)}^{{\sigma }_1}{(1\otimes \mathcal{V})}^{{\sigma }_2}={(1\otimes \mathcal{V})}^{{\sigma }_1}{(\mathcal{U}\otimes 1)}^{{\sigma }_2}={\mathcal{U}}^{{\sigma }_2}\otimes {\mathcal{V}}^{{\sigma }_1}.$$

Using the functional calculus and tensorial product characteristics for continuous functions of self-adjoint operators the associated tensorial perspective ${\mathcal{Q}}_{\zeta ,\otimes }$, is defined as follows:

$$\begin{array}{cc}\hfill {\mathcal{Q}}_{\zeta ,\otimes }(\mathcal{U},\mathcal{V})& =(1\otimes \mathcal{V})\zeta \left((\mathcal{U}\otimes 1){(1\otimes \mathcal{V})}^{-1}\right)=\zeta \left((\mathcal{U}\otimes 1){(1\otimes \mathcal{V})}^{-1}\right)(1\otimes \mathcal{V})\hfill \\ & =\zeta \left({(1\otimes \mathcal{V})}^{-1}(\mathcal{U}\otimes 1)\right)(1\otimes \mathcal{V}),\hfill \end{array}$$

due to the commutativity of $\mathcal{U}\otimes 1$ and $1\otimes \mathcal{V}$.

Lemma 1 

([33]). Let $\mathcal{U}$ and $\mathcal{V}$ be self-adjoint operators with $\mathcal{SP}\left(\mathcal{U}\right)\subset {∆}_1$ and $\mathcal{SP}\left(\mathcal{V}\right)\subset$${∆}_2$. If $\theta ,\zeta$ are continuous on ${∆}_1$, then $\Phi ,\Im$ are continuous on ${∆}_2$, and $\phi$ is continuous on $∆$, and subsequently the class of intervals $\zeta \left({∆}_1\right)+\Im \left({∆}_2\right)$ is

$$\begin{array}{cc}\hfill & (\theta (\mathcal{U})\otimes 1+1\otimes \Phi (\mathcal{V}\left)\right)\phi (\zeta (\mathcal{U})\otimes 1+1\otimes \Im (\mathcal{V}\left)\right)\hfill \\ \hfill & ={\int }_{{∆}_1}{\int }_{{∆}_2}(\theta (\sigma )+\Phi \left(\mathtt{u}\right))\phi (\zeta (\sigma )+\Im \left(\mathtt{u}\right))\mathtt{d}{\mathtt{E}}_{{∆}_1}\otimes {\mathtt{dF}}_{{∆}_2},\hfill \end{array}$$

(3)

where $\mathcal{U}$ and $\mathcal{V}$ are the spectral resolutions

$$\mathcal{U}={\int }_{{∆}_1}\alpha \mathtt{dE}(\alpha )\mathit{and}\mathcal{V}={\int }_{{∆}_2}\beta \mathtt{dF}(\beta ).$$

Definition 3 

([39]). A mapping $\zeta :\Delta \subseteq \mathsf{R}\to \mathsf{R}$ is stated to be convex (concave) on $\Delta$, if

$$\zeta (\alpha {\xi }_1+(1-\alpha ){\xi }_2)\le (\ge )\alpha \zeta \left({\xi }_1\right)+(1-\alpha )\zeta \left({\xi }_2\right),$$

valid for all ${\xi }_1,{\xi }_2\in \Delta$ and $\alpha \in [0,1]$.

Definition 4 

([39]). A mapping $\zeta :\Delta \to \mathsf{R}$ is stated to be convex in a quasi sense if

$$\zeta ((1-\alpha ){\xi }_1+\alpha {\xi }_2)\le max\{\zeta \left({\xi }_2\right),\zeta \left({\xi }_1\right)\}={\displaystyle \frac{1}{2}}(\zeta \left({\xi }_2\right)+\zeta \left({\xi }_1\right)+|\zeta \left({\xi }_2\right)-\zeta \left({\xi }_1\right)\left|\right),$$

for all ${\xi }_1,{\xi }_2\in \Delta$ and $\alpha \in [0,1]$.

Definition 5 

([39]). A mapping $\zeta :\Delta \subseteq \mathsf{R}\to \mathsf{R}$ is stated to be Godunova convex on $\Delta$, if

$$\zeta (\alpha {\xi }_1+(1-\alpha ){\xi }_2)\le {\displaystyle \frac{\zeta \left({\xi }_1\right)}{\alpha }}+{\displaystyle \frac{\zeta \left({\xi }_2\right)}{(1-\alpha )}},$$

valid for all ${\xi }_1,{\xi }_2\in \Delta$ and $\alpha \in (0,1)$.

The following lemma, whose properties are also utilized in our main findings, is presented in a similar manner using synchronous functions in the work of Dragomir [78].

Lemma 2. 

Assume that $\zeta ,\psi$ are asynchronous and continuous on $∆$, while $\phi ,\vartheta$ are continuous and non-negative on $∆$. If $\mathcal{U},\mathcal{V}$ are self-adjoint with spectrums $\mathcal{SP}\left(\mathcal{U}\right),\mathcal{SP}\left(\mathcal{V}\right)\subset ∆$, then

$$\begin{array}{cc}& [\phi (\mathcal{U})\zeta (\mathcal{U})\psi (\mathcal{U}\left)\right]\otimes \vartheta \left(\mathcal{V}\right)+\phi \left(\mathcal{U}\right)\otimes [\vartheta (\mathcal{V})\zeta (\mathcal{V})\psi (\mathcal{V}\left)\right]\hfill \\ & \le [\phi (\mathcal{U})\zeta (\mathcal{U}\left)\right]\otimes [\vartheta (\mathcal{V})\psi (\mathcal{V}\left)\right]+[\phi (\mathcal{U})\psi (\mathcal{U}\left)\right]\otimes [\vartheta (\mathcal{V})\zeta (\mathcal{V}\left)\right],\hfill \end{array}$$

or, equivalently

$$\begin{array}{cc}& (\phi (\mathcal{U})\otimes \vartheta (\mathcal{V}\left)\right)\left[\right(\zeta \left(\mathcal{U}\right)\psi \left(\mathcal{U}\right))\otimes 1+1\otimes (\zeta \left(\mathcal{V}\right)\psi \left(\mathcal{V}\right)\left)\right]\hfill \\ & \le (\phi (\mathcal{U})\otimes \vartheta (\mathcal{V}\left)\right)[\zeta (\mathcal{U})\otimes \psi (\mathcal{V})+\psi (\mathcal{U})\otimes \zeta (\mathcal{V}\left)\right].\hfill \end{array}$$

Proof. 

Suppose $\zeta$ and $\psi$ are asynchronous functions on ∆, then

$$\zeta (\tau )\psi (\tau )+\zeta \left(\upsilon \right)\psi \left(\upsilon \right)\le \zeta (\tau )\psi \left(\upsilon \right)+\zeta \left(\upsilon \right)\psi (\tau ),$$

for all $\tau ,\upsilon \in ∆$.

To obtain this inequality, we multiply it by $\phi (\tau )\vartheta \left(\upsilon \right)\le 0,$ and it follows that

$$\begin{array}{cc}& \zeta (\tau )\psi (\tau )\phi (\tau )\vartheta \left(\upsilon \right)+\phi (\tau )\zeta \left(\upsilon \right)\psi \left(\upsilon \right)\vartheta \left(\upsilon \right)\hfill \\ & \le \zeta (\tau )\phi (\tau )\psi \left(\upsilon \right)\vartheta \left(\upsilon \right)+\zeta \left(\upsilon \right)\vartheta \left(\upsilon \right)\psi (\tau )\phi (\tau ),\hfill \end{array}$$

for all $\tau ,\upsilon \in ∆$.

If we use the double integral, we obtain

$$\begin{array}{cc}& {\int }_{∆}{\int }_{∆}[\zeta (\tau )\psi (\tau )\phi (\tau )\vartheta \left(\upsilon \right)+\phi (\tau )\zeta \left(\upsilon \right)\psi \left(\upsilon \right)\vartheta \left(\upsilon \right)]\mathtt{d}\mathtt{E}(\tau )\otimes \mathtt{d}\mathtt{F}\left(\upsilon \right)\hfill \\ & \le {\int }_{∆}{\int }_{∆}[\zeta (\tau )\phi (\tau )\psi \left(\upsilon \right)\vartheta \left(\upsilon \right)+\zeta \left(\upsilon \right)\vartheta \left(\upsilon \right)\psi (\tau )\phi (\tau )]\mathtt{d}\mathtt{E}(\tau )\otimes \mathtt{d}\mathtt{F}\left(\upsilon \right).\hfill \end{array}$$

Note that

$$\begin{array}{cc}& {\int }_{∆}{\int }_{∆}[\zeta (\tau )\psi (\tau )\phi (\tau )\vartheta \left(\upsilon \right)+\phi (\tau )\zeta \left(\upsilon \right)\psi \left(\upsilon \right)\vartheta \left(\upsilon \right)]\mathtt{d}\mathtt{E}(\tau )\otimes \mathtt{d}\mathtt{F}\left(\upsilon \right)\hfill \\ & ={\int }_{∆}{\int }_{∆}\zeta (\tau )\psi (\tau )\phi (\tau )\vartheta \left(\upsilon \right)\mathtt{d}\mathtt{E}(\tau )\otimes \mathtt{d}\mathtt{F}\left(\upsilon \right)\hfill \\ & +{\int }_{∆}{\int }_{∆}\phi (\tau )\zeta \left(\upsilon \right)\psi \left(\upsilon \right)\vartheta \left(\upsilon \right)\mathtt{d}\mathtt{E}(\tau )\otimes \mathtt{d}\mathtt{F}\left(\upsilon \right)\hfill \\ & =[\phi (\mathcal{U})\zeta (\mathcal{U})\psi (\mathcal{U}\left)\right]\otimes \vartheta \left(\mathcal{V}\right)+\phi \left(\mathcal{U}\right)\otimes [\vartheta (\mathcal{V})\zeta (\mathcal{V})\psi (\mathcal{V}\left)\right]\hfill \end{array}$$

and

$$\begin{array}{cc}& {\int }_{∆}{\int }_{∆}[\zeta (\tau )\phi (\tau )\psi \left(\upsilon \right)\vartheta \left(\upsilon \right)+\zeta \left(\upsilon \right)\vartheta \left(\upsilon \right)\psi (\tau )\phi (\tau )]\mathtt{d}\mathtt{E}(\tau )\otimes \mathtt{d}\mathtt{F}\left(\upsilon \right)\hfill \\ & ={\int }_{∆}{\int }_{∆}\zeta (\tau )\phi (\tau )\psi \left(\upsilon \right)\vartheta \left(\upsilon \right)\mathtt{d}\mathtt{E}(\tau )\otimes \mathtt{d}\mathtt{F}\left(\upsilon \right)\hfill \\ & +{\int }_{∆}{\int }_{∆}\psi (\tau )\phi (\tau )\zeta \left(\upsilon \right)\vartheta \left(\upsilon \right)\mathtt{d}\mathtt{E}(\tau )\otimes \mathtt{d}\mathtt{F}\left(\upsilon \right)\hfill \\ & =[\phi (\mathcal{U})\zeta (\mathcal{U}\left)\right]\otimes [\vartheta (\mathcal{V})\psi (\mathcal{V}\left)\right]+[\phi (\mathcal{U})\psi (\mathcal{U}\left)\right]\otimes [\vartheta (\mathcal{V})\zeta (\mathcal{V}\left)\right].\hfill \end{array}$$

Now, by applying the tensorial property

$$\left(\mathtt{PQ}\right)\otimes \left(\mathtt{RS}\right)=(\mathtt{P}\otimes \mathtt{R})(\mathtt{Q}\otimes \mathtt{u}),$$

for any $\mathtt{P},\mathtt{Q},\mathtt{R},\mathtt{S}\in \mathcal{B}\left(\mathcal{H}\right)$, we obtain

$$\begin{array}{cc}& [\phi (\mathcal{U})\zeta (\mathcal{U})\psi (\mathcal{U}\left)\right]\otimes \vartheta \left(\mathcal{V}\right)+\phi \left(\mathcal{U}\right)\otimes [\vartheta (\mathcal{V})\zeta (\mathcal{V})\psi (\mathcal{V}\left)\right]\hfill \\ & =(\phi (\mathcal{U})\otimes \vartheta (\mathcal{V}\left)\right)\left[\right(\zeta \left(\mathcal{U}\right)\psi \left(\mathcal{U}\right))\otimes 1]+(\phi (\mathcal{U})\otimes \vartheta (\mathcal{V}\left)\right)[1\otimes (\zeta \left(\mathcal{V}\right)\psi \left(\mathcal{V}\right)\left)\right]\hfill \\ & =(\phi (\mathcal{U})\otimes \vartheta (\mathcal{V}\left)\right)\left[\right(\zeta \left(\mathcal{U}\right)\psi \left(\mathcal{U}\right))\otimes 1+1\otimes (\zeta \left(\mathcal{V}\right)\psi \left(\mathcal{V}\right)\left)\right]\hfill \end{array}$$

and

$$\begin{array}{cc}& [\phi (\mathcal{U})\zeta (\mathcal{U}\left)\right]\otimes [\vartheta (\mathcal{V})\psi (\mathcal{V}\left)\right]+[\phi (\mathcal{U})\psi (\mathcal{U}\left)\right]\otimes [\vartheta (\mathcal{V})\zeta (\mathcal{V}\left)\right]\hfill \\ & =(\phi (\mathcal{U})\otimes \vartheta (\mathcal{V}\left)\right)(\zeta (\mathcal{U})\otimes \psi (\mathcal{V}\left)\right)+(\phi (\mathcal{U})\otimes \vartheta (\mathcal{V}\left)\right)(\psi (\mathcal{U})\otimes \zeta (\mathcal{V}\left)\right)\hfill \\ & =(\phi (\mathcal{U})\otimes \vartheta (\mathcal{V}\left)\right)[\zeta (\mathcal{U})\otimes \psi (\mathcal{V})+\psi (\mathcal{U})\otimes \zeta (\mathcal{V}\left)\right],\hfill \end{array}$$

which proves Lemma 2. □

3. The Major Results

These two results play an important role in our further subsequent findings related to Macalurin approximation equality.

Lemma 3. 

Suppose that $\mathcal{U}$ and $\mathcal{V}$ are self-adjoint operators such that $\mathcal{SP}\left(\mathcal{U}\right)\subset {∆}_1$ and $\mathcal{SP}\left(\mathcal{V}\right)\subset$${∆}_2$. If $\zeta ,\phi$ are continuous on ${∆}_1,\psi ,\vartheta$ are continuous on ${∆}_2$ and $\xi$, $\eta$ are continuous on interval $\mathcal{K}$, subsequently the sum of $\zeta \left({∆}_1\right)+\psi \left({∆}_2\right),\phi \left({∆}_1\right)+\vartheta \left({∆}_2\right)\subset \mathcal{K}$ is

$$\begin{array}{cc}& \xi (\zeta (\mathcal{U})\otimes 1+1\otimes \psi (\mathcal{V}\left)\right)\eta (\phi (\mathcal{U})\otimes 1+1\otimes \vartheta (\mathcal{V}\left)\right)\hfill \\ & ={\int }_{{∆}_1}{\int }_{{∆}_2}\xi (\zeta (\tau )+\psi \left(\upsilon \right))\eta (\phi (\tau )+\vartheta \left(\upsilon \right)){\mathtt{d}\mathtt{E}}_{\upsilon }\otimes {\mathtt{d}\mathtt{F}}_{\tau }.\hfill \end{array}$$

Proof. 

Let $\mathtt{a},\mathtt{b},\mathtt{c}$ and $\mathtt{d}$ positive-continuous functions such that $\zeta (\tau )=ln\mathtt{a}(\tau )$, $\phi (\tau )=ln\mathtt{c}(\tau )$ for $\tau \in {∆}_1$ and $\psi \left(\upsilon \right)=ln\mathtt{b}(\mathtt{\upsilon }),\vartheta \left(\upsilon \right)=ln\mathtt{d}(\mathtt{\upsilon })$ for $\upsilon \in {∆}_2$. Then,

$$\begin{array}{cc}& {\int }_{{∆}_1}{\int }_{{∆}_2}\xi (\zeta (\tau )+\psi \left(\upsilon \right))\eta (\phi (\tau )+\vartheta \left(\upsilon \right)){\mathtt{d}\mathtt{E}}_{\upsilon }\otimes {\mathtt{d}\mathtt{F}}_{\tau }\hfill \\ & ={\int }_{{∆}_1}{\int }_{{∆}_2}\xi (ln\mathtt{a}(\tau )+ln\mathtt{b}(\mathtt{\upsilon }))\eta (ln\mathtt{c}(\tau )+ln\mathtt{d}(\mathtt{\upsilon })){\mathtt{d}\mathtt{E}}_{\upsilon }\otimes {\mathtt{d}\mathtt{F}}_{\tau }\hfill \\ & ={\int }_{{∆}_1}{\int }_{{∆}_2}(\xi \circ ln)\left(\mathtt{a}(\tau )\mathtt{b}(\mathtt{\upsilon })\right)(\eta \circ ln)\left(\mathtt{c}(\tau )\mathtt{d}(\mathtt{\upsilon })\right){\mathtt{d}\mathtt{E}}_{\upsilon }\otimes {\mathtt{d}\mathtt{F}}_{\tau }.\hfill \end{array}$$

By applying Lemma 2 to the functions $(\xi \circ ln)$ and $(\eta \circ ln)$, we obtain

$$\begin{array}{cc}& {\int }_{{∆}_1}{\int }_{{∆}_2}(\xi \circ ln)\left(\mathtt{a}(\tau )\mathtt{b}(\mathtt{\upsilon })\right)(\eta \circ ln)\left(\mathtt{c}(\tau )\mathtt{d}(\mathtt{\upsilon })\right){\mathtt{d}\mathtt{E}}_{\upsilon }\otimes {\mathtt{d}\mathtt{F}}_{\tau }\hfill \\ & =(\xi \circ ln)\left(\mathtt{a}\right(\mathcal{U})\otimes \mathtt{b}(\mathcal{V}\left)\right)(\eta \circ ln)\left(\mathtt{c}\right(\mathcal{U})\otimes \mathtt{d}(\mathcal{V}\left)\right)\hfill \\ & =\xi [ln(\mathtt{a}\left(\mathcal{U}\right)\otimes \mathtt{b}\left(\mathcal{V}\right)\left)\right]\eta [ln(\mathtt{c}\left(\mathcal{U}\right)\otimes \mathtt{d}\left(\mathcal{V}\right)\left)\right].\hfill \end{array}$$

Now, observe that, by the commutativity of the operators $\mathtt{a}\left(\mathcal{U}\right)\otimes 1$ and $1\otimes \mathtt{b}\left(\mathcal{V}\right)$,

$$\begin{array}{cc}\hfill ln\left(\mathtt{a}\right(\mathcal{U})\otimes \mathtt{b}(\mathcal{V}\left)\right)& =ln\left[\right(\mathtt{a}\left(\mathcal{U}\right)\otimes 1\left)\right(1\otimes \mathtt{b}\left(\mathcal{V}\right)\left)\right]\hfill \\ & =ln\left(\mathtt{a}\right(\mathcal{U})\otimes 1)+ln(1\otimes \mathtt{b}(\mathcal{V}\left)\right)\hfill \\ & =[ln\mathtt{a}(\mathcal{U}\left)\right]\otimes 1+1\otimes ln\mathtt{b}\left(\mathcal{V}\right)\hfill \\ & =\zeta \left(\mathcal{U}\right)\otimes 1+1\otimes \psi \left(\mathcal{V}\right)\hfill \end{array}$$

and, similarly

$$ln\left(\mathtt{c}\right(\mathcal{U})\otimes \mathtt{d}(\mathcal{V}\left)\right)=\phi \left(\mathcal{U}\right)\otimes 1+1\otimes \vartheta \left(\mathcal{V}\right).$$

Hence, we obtain the desired representation for Lemma 3. □

Theorem 7. 

If $\zeta$ is a Godunova–Levin mapping defined over the positive half-line, then ${\mathcal{Q}}_{\zeta ,\otimes }$ is also Godunova–Levin in pairs of positive definite operators. If $\mathcal{U}\ge \mathcal{G}>0$ and $\mathcal{V}\ge \mathcal{H}>0$, then also

$${\mathcal{Q}}_{\zeta ,\otimes }(\mathcal{U},\mathcal{V})\ge {\mathcal{Q}}_{\zeta ,\otimes }(\mathcal{G},\mathcal{H}).$$

Proof. 

Suppose that $\zeta$ is a operator Godunova–Levin function in the positive half-line. Given the commutative nature of $\mathcal{U}\otimes 1$ and $1\otimes 0$, we have

$${\mathcal{Q}}_{\zeta ,\otimes }(\mathcal{U},\mathcal{V})=(1\otimes \mathcal{V})\zeta \left((\mathcal{U}\otimes 1){(1\otimes \mathcal{V})}^{-1}\right)={\mathcal{Q}}_{\zeta }(\mathcal{U}\otimes 1,1\otimes \mathcal{V})$$

for $\mathcal{U},\mathcal{V}>0$. If $\mathcal{U},\mathcal{V},\mathcal{G},\mathcal{H}>0$ and $\alpha \in (0,1)$, then we have

$$\begin{array}{cc}& {\mathcal{Q}}_{\zeta },\otimes ((1-\alpha )(\mathcal{U},\mathcal{V})+\alpha (\mathcal{G},\mathcal{H}))\hfill \\ & ={\mathcal{Q}}_{\zeta },\otimes \left(((1-\alpha )\mathcal{U}+\alpha \mathcal{G},(1-\alpha )\mathcal{V}+\alpha \mathcal{H})\right)\hfill \\ & ={\mathcal{Q}}_{\zeta }(((1-\alpha )\mathcal{U}+\alpha \mathcal{G})\otimes 1,1\otimes ((1-\alpha )\mathcal{V}+\alpha \mathcal{H}))\hfill \\ & ={\mathcal{Q}}_{\zeta }((1-\alpha )\mathcal{U}\otimes 1+\alpha \mathcal{G}\otimes 1,(1-\alpha )1\otimes \mathcal{V}+\alpha 1\otimes \mathcal{H})\hfill \\ & ={\mathcal{Q}}_{\zeta }((1-\alpha )(\mathcal{U}\otimes 1,1\otimes \mathcal{V})+\alpha (\mathcal{G}\otimes 1,1\otimes \mathcal{H}))\hfill \\ & \le {\displaystyle \frac{{\mathcal{Q}}_{\zeta }(\mathcal{U}\otimes 1,1\otimes \mathcal{V})}{(1-\alpha )}}+{\displaystyle \frac{{\mathcal{Q}}_{\zeta }(\mathcal{G}\otimes 1,1\otimes \mathcal{H})}{\alpha }}\hfill \\ & ={\displaystyle \frac{{\mathcal{Q}}_{\zeta },\otimes (\mathcal{U},\mathcal{V})}{(1-\alpha )}}+{\displaystyle \frac{{\mathcal{Q}}_{\zeta },\otimes (\mathcal{G},\mathcal{H})}{\alpha }},\hfill \end{array}$$

and it demonstrates that ${\mathcal{Q}}_{\zeta ,\otimes }$ is a Godunova–Levin operator in pairs of positive definite operators. If $\mathcal{U}\ge \mathcal{G}>0$ and $\mathcal{V}\ge \mathcal{H}>0$, then $\mathcal{U}\otimes 1\ge \mathcal{G}\otimes 1>0$ and $1\otimes 0\ge 1\otimes \mathcal{H}>0$, so we have

$${\mathcal{Q}}_{\zeta }(\mathcal{U}\otimes 1,1\otimes 0)\le {\mathcal{Q}}_{\zeta }(\mathcal{G}\otimes 1,1\otimes \mathcal{H}).$$

□

Tensorial Maclaurin Approximation Using Different Types of Generalize Convex Mappings

Lemma 4. 

Assume that $\zeta$ is of class ${\mathcal{C}}^{\mathtt{n}+1}$ on the open interval $∆$. Let $\mathcal{U}$ and $\mathcal{V}$ be self-adjoint operators with respective spectrums, $\mathcal{SP}\left(\mathcal{U}\right)\subset ∆$ and $\mathcal{SP}\left(\mathcal{V}\right)\subset ∆$, then one has

$$\begin{array}{cc}\hfill \zeta \left(\mathcal{U}\right)\otimes 1& =\sum _{\mathtt{k}=0}^{\mathtt{n}}{\displaystyle \frac{1}{\mathtt{k}!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{\mathtt{k}}\left(1\otimes {\zeta }^{\left(\mathtt{k}\right)}\left(0\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathtt{n}!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{\mathtt{n}+\mathtt{1}}\hfill \\ \hfill & \times {\int }_0^1{(\mathtt{1}-\mathtt{u})}^{\mathtt{n}}{\zeta }^{(\mathtt{n}+\mathtt{1})}((\mathtt{1}-\mathtt{u})1\otimes 0+\mathtt{u}\mathcal{U}\otimes 1)\mathtt{d}\mathtt{u}\hfill \end{array}$$

(4)

Proof. 

Using Maclaurin’s representation with the integral remainder, we can express the following two identities:

$$\zeta (\mu )=\sum _{\mathtt{k}=0}^{\mathtt{n}}{\displaystyle \frac{1}{\mathtt{k}!}}{\zeta }^{\left(\mathtt{k}\right)}\left(\mathcal{U}\right){(\mu -0)}^{\mathtt{k}}+{\displaystyle \frac{1}{\mathtt{n}!}}{\int }_0^{\mu }{\zeta }^{(\mathtt{n}+\mathtt{1})}(\nu ){(\mu -\nu )}^{\mathtt{n}}\mathtt{d}\nu$$

for $\mu \in ∆$.

Using the substitution $\nu =(\mathtt{1}-\mathtt{u}){\sigma }_1+\mathtt{u}{\sigma }_2,\mathtt{u}\in [0,1]$ and for any continous function $\phi$ with distinct values ${\sigma }_1,{\sigma }_2$ in that interval, one has

$${\int }_{{\sigma }_1}^{{\sigma }_2}\phi (\nu )\mathtt{d}\nu =({\sigma }_2-{\sigma }_1){\int }_0^1\phi ((\mathtt{1}-\mathtt{u}){\sigma }_1+\mathtt{u}{\sigma }_2)\mathtt{du}.$$

Therefore,

$$\begin{array}{cc}& {\int }_0^{\mu }{\zeta }^{(\mathtt{n}+\mathtt{1})}(\nu ){(\mu -\nu )}^{\mathtt{n}}\mathtt{d}\nu \hfill \\ & =(\mu -0){\int }_0^1{\zeta }^{(\mathtt{n}+\mathtt{1})}((\mathtt{1}-\mathtt{u})0+\mathtt{u}\mathtt{x}){(\mathtt{x}-(\mathtt{1}-\mathtt{u})0-\mathtt{u}\mathtt{x})}^{\mathtt{n}}\mathtt{d}\mathtt{u}\hfill \\ & ={(\mu -0)}^{\mathtt{n}+\mathtt{1}}{\int }_0^1{\zeta }^{(\mathtt{n}+\mathtt{1})}((\mathtt{1}-\mathtt{u})0+\mathtt{u}x){(\mathtt{1}-\mathtt{u})}^{\mathtt{n}}\mathtt{d}\mathtt{u}\hfill \end{array}$$

and it follows that

$$\begin{array}{cc}\hfill \zeta (\tau )& =\sum _{\mathtt{k}=0}^{\mathtt{n}}{\displaystyle \frac{1}{\mathtt{k}!}}{\zeta }^{\left(\mathtt{k}\right)}\left(0\right){(\tau -0)}^{\mathtt{k}}\hfill \\ & +{\displaystyle \frac{1}{\mathtt{n}!}}{(\tau -0)}^{\mathtt{n}+\mathtt{1}}{\int }_0^1{\zeta }^{(\mathtt{n}+\mathtt{1})}((\mathtt{1}-\mathtt{u})0+\mathtt{u}\tau ){(\mathtt{1}-\mathtt{u})}^{\mathtt{n}}\mathtt{d}\mathtt{u}\hfill \end{array}$$

for all $\tau ,\upsilon \in ∆$. If $\mathcal{U}$ and $\mathcal{V}$ have the spectral resolutions

$$\mathcal{U}={\int }_{∆}\tau \mathtt{dE}(\tau )\mathrm{and}\mathcal{V}={\int }_{∆}\upsilon \mathtt{dF}\left(\upsilon \right),$$

and after that, we obtain the integral applying ${\int }_{∆}{\int }_{∆}$ over ${\mathtt{dF}}_{\tau }\otimes {\mathtt{dF}}_{\upsilon }$, that is

$$\begin{array}{cc}& {\int }_{∆}{\int }_{∆}\zeta (\tau ){\mathtt{dF}}_{\tau }\otimes {\mathtt{dF}}_{\upsilon }\hfill \\ & ={\int }_{∆}{\int }_{∆}\left(\sum _{\mathtt{k}=0}^{\mathtt{n}}{\displaystyle \frac{1}{\mathtt{k}!}}{\zeta }^{\left(\mathtt{k}\right)}\left(0\right){(\tau -0)}^{\mathtt{k}}\right){\mathtt{dF}}_{\tau }\otimes {\mathtt{dF}}_{\upsilon }\hfill \\ & +{\displaystyle \frac{1}{\mathtt{n}!}}{\int }_{∆}{\int }_{∆}{(\tau -0)}^{\mathtt{n}+\mathtt{1}}\left({\int }_0^1{\zeta }^{(\mathtt{n}+\mathtt{1})}((\mathtt{1}-\mathtt{u})0+\mathtt{u}\tau ){(\mathtt{1}-\mathtt{u})}^{\mathtt{n}}\mathtt{d}\mathtt{u}\right){\mathtt{dF}}_{\tau }\otimes {\mathtt{dF}}_{\upsilon }.\hfill \end{array}$$

We have

$${\int }_{∆}{\int }_{∆}\zeta (\tau ){\mathtt{dF}}_{\tau }\otimes {\mathtt{dF}}_{\upsilon }=\zeta \left(\mathcal{U}\right)\otimes 1$$

and, by Lemma 3, we have

$$\begin{array}{cc}& {\int }_{∆}{\int }_{∆}\left(\sum _{\mathtt{k}=0}^{\mathtt{n}}{\displaystyle \frac{1}{\mathtt{k}!}}{\zeta }^{\left(\mathtt{k}\right)}\left(0\right){(\tau -0)}^{\mathtt{k}}\right){\mathtt{dF}}_{\tau }\otimes {\mathtt{dF}}_{\upsilon }\hfill \\ & =\sum _{\mathtt{k}=0}^{\mathtt{n}}{\displaystyle \frac{1}{\mathtt{k}!}}{\int }_{∆}{\int }_{∆}\left({(\tau -0)}^{\mathtt{k}}{\zeta }^{\left(\mathtt{k}\right)}\left(0\right)\right){\mathtt{dF}}_{\tau }\otimes {\mathtt{dF}}_{\upsilon }\hfill \\ & =\sum _{\mathtt{k}=0}^{\mathtt{n}}{\displaystyle \frac{1}{\mathtt{k}!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{\mathtt{k}}\left(1\otimes {\zeta }^{\left(\mathtt{k}\right)}\left(0\right)\right).\hfill \end{array}$$

Using Fubini’s theorem, we obtain

$$\begin{array}{cc}& {\int }_{∆}{\int }_{∆}{(\tau -0)}^{\mathtt{n}+\mathtt{1}}\left({\int }_0^1{\zeta }^{(\mathtt{n}+\mathtt{1})}((\mathtt{1}-\mathtt{u})0+\mathtt{u}\tau ){(\mathtt{1}-\mathtt{u})}^{\mathtt{n}}\mathtt{d}\mathtt{u}\right){\mathtt{dF}}_{\tau }\otimes {\mathtt{dF}}_{\upsilon }\hfill \\ & ={\int }_0^1{(\mathtt{1}-\mathtt{u})}^{\mathtt{n}}\left({\int }_{∆}{\int }_{∆}{(\tau -0)}^{\mathtt{n}+\mathtt{1}}\left({\zeta }^{(\mathtt{n}+\mathtt{1})}((\mathtt{1}-\mathtt{u})0+\mathtt{u}\tau )\right){\mathtt{dF}}_{\tau }\otimes {\mathtt{dF}}_{\upsilon }\right)\mathtt{d}\mathtt{u}\hfill \\ & ={(\mathcal{U}\otimes 1-1\otimes 0)}^{\mathtt{n}+\mathtt{1}}{\int }_0^1{(\mathtt{1}-\mathtt{u})}^{\mathtt{n}}\left({\zeta }^{(\mathtt{n}+\mathtt{1})}((\mathtt{1}-\mathtt{u})1\otimes 0+\mathtt{u}\mathcal{U}\otimes 1)\right)\mathtt{d}\mathtt{u},\hfill \end{array}$$

and so we obtain the required outcome. □

Theorem 8. 

Assume that $\zeta$ is of class ${\mathcal{C}}^{2\wp +1}$ on the open interval $∆$ such that ${\alpha }_{2\wp +1}\le {\zeta }^{(2\wp +1)}\le {\beta }_{2\wp +1}$ for some constants ${\alpha }_{2\wp +1},{\beta }_{2\wp +1}.$ Let $\mathcal{U}$ and $\mathcal{V}$ be self-adjoint operators with respective spectrums, $\mathcal{SP}\left(\mathcal{U}\right)\subset ∆$ and $\mathcal{SP}\left(\mathcal{V}\right)\subset ∆$, then one has

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{(2\wp +1)!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}{\alpha }_{2\wp +1}\hfill \\ \hfill & \le \zeta \left(\mathcal{U}\right)\otimes 1-\sum _{\mathtt{k}=0}^{2\wp }{\displaystyle \frac{1}{\mathtt{k}!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{\mathtt{k}}\left(1\otimes {\zeta }^{\left(\mathtt{k}\right)}\left(0\right)\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{(2\wp +1)!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}{\beta }_{2\wp +1}.\hfill \end{array}$$

(5)

Proof. 

For $\mathtt{n}=2\wp$ with $\wp \ge 1$ in (4), we have

$$\begin{array}{cc}\hfill & \zeta \left(\mathcal{U}\right)\otimes 1\hfill \\ \hfill & =\sum _{\mathtt{k}=0}^{2\wp }{\displaystyle \frac{1}{\mathtt{k}!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{\mathtt{k}}\left(1\otimes {\zeta }^{\left(\mathtt{k}\right)}\left(0\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{(2\wp )!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\hfill \\ \hfill & \times {\int }_0^1{(\mathtt{1}-\mathtt{u})}^{2\wp }{\zeta }^{(2\wp +1)}((\mathtt{1}-\mathtt{u})1\otimes 0+\mathtt{u}\mathcal{U}\otimes 1)\mathtt{d}\mathtt{u}.\hfill \end{array}$$

(6)

Based on the boundedness assumption, we have that

$$\begin{array}{c}\hfill {\alpha }_{2\wp +1}\le {\zeta }^{\left(2\wp +1\right)}((\mathtt{1}-\mathtt{u})0+\mathtt{ut})\le {\beta }_{2\wp +1},\end{array}$$

(7)

for all $\mathtt{t}\in ∆$ and $\mathtt{u}\in [0,1]$.

Multiplying the inequality (7) by ${(\mathtt{t}-0)}^{2\wp +1}\ge 0$ with $\mathtt{t}\in ∆$ yields

$$\begin{array}{c}\hfill {\alpha }_{2\wp +1}{(\mathtt{t}-0)}^{2\wp +1}\le {(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{\left(2\wp +1\right)}((\mathtt{1}-\mathtt{u})0+\mathtt{ut})\le {\beta }_{2\wp +1}{(\mathtt{t}-0)}^{2\wp +1}.\end{array}$$

(8)

By taking the integral ${\int }_{∆}{\int }_{∆}$ over ${\mathtt{dE}}_{\mathtt{t}}\otimes {\mathtt{dF}}_{\mathtt{s}}$ in (8), we obtain

$$\begin{array}{cc}& {\alpha }_{2\wp +1}{\int }_{∆}{\int }_{∆}{(\mathtt{t}-0)}^{2\wp +1}\mathtt{d}{\mathtt{E}}_{\mathtt{t}}\otimes \mathtt{d}{\mathtt{F}}_{\mathtt{s}}\hfill \\ & \le {\int }_{∆}{\int }_{∆}{(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{\left(2\wp +1\right)}((\mathtt{1}-\mathtt{u})0+\mathtt{ut})\mathtt{d}{\mathtt{E}}_{\mathtt{t}}\otimes \mathtt{d}{\mathtt{F}}_{\mathtt{s}}\hfill \\ & \le {\beta }_{2\wp +1}{\int }_{∆}{\int }_{∆}{(\mathtt{t}-0)}^{2\wp +1}\mathtt{d}{\mathtt{E}}_{\mathtt{t}}\otimes \mathtt{d}{\mathtt{F}}_{\mathtt{s}}.\hfill \end{array}$$

Using Lemma 4, we derive

$$\begin{array}{cc}& {\alpha }_{2\wp +1}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\hfill \\ & \le {(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}{\zeta }^{\left(2\wp +1\right)}((\mathtt{1}-\mathtt{u})1\otimes 0+\mathtt{u}\mathcal{U}\otimes 1)\hfill \\ & \le {\beta }_{2\wp +1}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}.\hfill \end{array}$$

In addition, if we multiply this inequality by ${(1-\mathtt{u})}^{2\wp +1}\ge 0$, $\mathtt{u}\in [0,1]$ and integrate, we obtain

$$\begin{array}{cc}& {\alpha }_{2\wp +1}{\int }_0^1{(\mathtt{1}-\mathtt{u})}^{2\wp +1}\mathtt{d}\mathtt{u}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\hfill \\ & \le {\int }_0^1{(\mathtt{1}-\mathtt{u})}^{2\wp +1}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}{\zeta }^{\left(2\wp +1\right)}((\mathtt{1}-\mathtt{u})1\otimes 0+\mathtt{u}\mathcal{U}\otimes 1)\mathtt{d}\mathtt{u}\hfill \\ & \le {\beta }_{2\wp +1}{\int }_0^1{(\mathtt{1}-\mathtt{u})}^{2\wp +1}\mathtt{d}\mathtt{u}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1},\hfill \end{array}$$

which gives that

$$\begin{array}{cc}& {\displaystyle \frac{{\alpha }_{2\wp +1}}{2\wp +1}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\hfill \\ & \le {(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}{\int }_0^1{(\mathtt{1}-\mathtt{u})}^{2\wp +1}{\zeta }^{\left(2\wp +1\right)}((\mathtt{1}-\mathtt{u})1\otimes 0+\mathtt{u}\mathcal{U}\otimes 1)\mathtt{d}\mathtt{u}\hfill \\ & \le {\displaystyle \frac{{\beta }_{2\wp +1}}{2\wp +1}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1},\hfill \end{array}$$

namely

$$\begin{array}{cc}& {\displaystyle \frac{{\alpha }_{2\wp +1}}{\left(2\wp +1\right)!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\hfill \\ & \le {\displaystyle \frac{1}{(2\wp )!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\times {\int }_0^1{(\mathtt{1}-\mathtt{u})}^{2\wp +1}{\zeta }^{\left(2\wp +1\right)}((\mathtt{1}-\mathtt{u})1\otimes 0+\mathtt{u}\mathcal{U}\otimes 1)\mathtt{d}\mathtt{u}\hfill \\ & \le {\displaystyle \frac{{\beta }_{2\wp +1}}{\left(2\wp +1\right)!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}.\hfill \end{array}$$

By utilizing identity (6), we derive the desired inequality. □

Theorem 9. 

Assume that $\zeta$ is of class ${\mathcal{C}}^{2\wp +1}$ on the open interval $∆$ such that ${\zeta }^{2\wp +1}$ is a convex function on $∆$. Let $\mathcal{U}$ and $\mathcal{V}$ be self-adjoint operators with respective spectrums, $\mathcal{SP}\left(\mathcal{U}\right)\subset ∆$ and $\mathcal{SP}\left(\mathcal{V}\right)\subset ∆$, then one has

$$\begin{array}{cc}\hfill & \zeta \left(\mathcal{U}\right)\otimes 1-\sum _{\mathtt{k}=0}^{2\wp }{\displaystyle \frac{1}{\mathtt{k}!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{\mathtt{k}}\left(1\otimes {\zeta }^{\left(\mathtt{k}\right)}\left(0\right)\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{(2\wp +1)!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\left[{\displaystyle \frac{2\wp +1\left(1\otimes {\zeta }^{(2\wp +1)}\left(0\right)\right)+{\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1}{2\wp +2}}\right].\hfill \end{array}$$

(9)

Proof. 

Since ${\zeta }^{(2\wp +1)}$ is convex on ∆, then

$$\begin{array}{c}\hfill {\zeta }^{(2\wp +1)}((\mathtt{1}-\mathtt{u})0+\mathtt{ut})\le (\mathtt{1}-\mathtt{u}){\zeta }^{(2\wp +1)}\left(0\right)+\mathtt{u}{\zeta }^{(2\wp +1)}\left(\mathtt{t}\right),\end{array}$$

(10)

for all $\mathtt{t}\in ∆$ and $\mathtt{u}\in [0,1]$.

Now, if we multiply the above inequality by ${(\mathtt{t}-0)}^{2\wp +1}\ge 0$ with $\mathtt{t}\in ∆$, then we have

$$\begin{array}{cc}\hfill & {(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{(2\wp +1)}((\mathtt{1}-\mathtt{u})0+\mathtt{ut})\hfill \\ \hfill & \le (\mathtt{1}-\mathtt{u}){(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{(2\wp +1)}\left(0\right)+\mathtt{u}{(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{(2\wp +1)}\left(\mathtt{t}\right),\hfill \end{array}$$

(11)

for all $\mathtt{t}\in ∆$ and $\mathtt{u}\in [0,1]$.

By applying the integral ${\int }_{∆}{\int }_{∆}$ over ${\mathtt{dE}}_{\mathtt{t}}\otimes {\mathtt{dF}}_{\mathtt{s}}$ in (11), we obtain

$$\begin{array}{cc}\hfill & {\int }_{∆}{\int }_{∆}{(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{(2\wp +1)}((\mathtt{1}-\mathtt{u})0+\mathtt{ut})\mathtt{d}{\mathtt{E}}_{\mathtt{t}}\otimes \mathtt{d}{\mathtt{F}}_{\mathtt{s}}\hfill \\ \hfill & \le {\int }_{∆}{\int }_{∆}\left[(\mathtt{1}-\mathtt{u}){(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{(2\wp +1)}\left(0\right)+\mathtt{u}{(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{(2\wp +1)}\left(\mathtt{t}\right)\right]\mathtt{d}{\mathtt{E}}_{\mathtt{t}}\otimes \mathtt{d}{\mathtt{F}}_{\mathtt{s}},\hfill \end{array}$$

(12)

for all $\mathtt{u}\in [0,1]$. Using Lemma 4, we conclude that

$$\begin{array}{cc}& {\int }_{∆}{\int }_{∆}{(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{(2\wp +1)}((\mathtt{1}-\mathtt{u})0+\mathtt{ut})\mathtt{d}{\mathtt{E}}_{\mathtt{t}}\otimes \mathtt{d}{\mathtt{F}}_{\mathtt{s}}\hfill \\ & ={(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}{\zeta }^{(2\wp +1)}((\mathtt{1}-\mathtt{u})1\otimes 0+\mathtt{u}\mathcal{U}\otimes 1)\hfill \end{array}$$

and

$$\begin{array}{cc}& {\int }_{∆}{\int }_{∆}\left[(\mathtt{1}-\mathtt{u}){(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{(2\wp +1)}\left(0\right)+\mathtt{u}{(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{(2\wp +1)}\left(\mathtt{t}\right)\right]\mathtt{d}{\mathtt{E}}_{\mathtt{t}}\otimes \mathtt{d}{\mathtt{F}}_{\mathtt{s}}\hfill \\ & =(\mathtt{1}-\mathtt{u}){\int }_{∆}{\int }_{∆}{(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{(2\wp +1)}\left(0\right)\mathtt{d}{\mathtt{E}}_{\mathtt{t}}\otimes \mathtt{d}{\mathtt{F}}_{\mathtt{s}}\hfill \\ & +\mathtt{u}{\int }_{∆}{\int }_{∆}{(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{(2\wp +1)}\left(\mathtt{t}\right)\mathtt{d}{\mathtt{E}}_{\mathtt{t}}\otimes \mathtt{d}{\mathtt{F}}_{\mathtt{s}}\hfill \\ & =(\mathtt{1}-\mathtt{u}){(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\left(1\otimes {\zeta }^{(2\wp +1)}\left(0\right)\right)\hfill \\ & +\mathtt{u}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\left({\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1\right)\hfill \end{array}$$

and by (12), we obtain

$$\begin{array}{cc}& {(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}{\zeta }^{(2\wp +1)}((\mathtt{1}-\mathtt{u})1\otimes 0+\mathtt{u}\mathcal{U}\otimes 1)\hfill \\ & \le (\mathtt{1}-\mathtt{u}){(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\left(1\otimes {\zeta }^{(2\wp +1)}\left(0\right)\right)\hfill \\ & +\mathtt{u}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\left({\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1\right),\hfill \end{array}$$

for all $\mathtt{u}\in [0,1]$. If we multiply the above inequality by ${(\mathtt{1}-\mathtt{u})}^{2\wp }$ and integrate, then we have

$$\begin{array}{cc}& {(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}{\int }_0^1{(\mathtt{1}-\mathtt{u})}^{2\wp }{\zeta }^{(2\wp +1)}((\mathtt{1}-\mathtt{u})1\otimes 0+\mathtt{u}\mathcal{U}\otimes 1)\mathtt{d}\mathtt{u}\hfill \\ & \le {\int }_0^1{(\mathtt{1}-\mathtt{u})}^{2\wp +1}\mathtt{d}\mathtt{u}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\left(1\otimes {\zeta }^{(2\wp +1)}\left(0\right)\right)\hfill \\ & +{\int }_0^1\mathtt{u}{(\mathtt{1}-\mathtt{u})}^{2\wp }\mathtt{d}\mathtt{u}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\left({\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1\right)\hfill \\ & ={\displaystyle \frac{1}{2\wp +2}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\left(1\otimes {\zeta }^{(2\wp +1)}\left(0\right)\right)\hfill \\ & +{\displaystyle \frac{1}{2\wp (2\wp +2)}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\left({\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1\right)\hfill \\ & ={\displaystyle \frac{1}{2\wp +2}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\left[1\otimes {\zeta }^{(2\wp +1)}0)+{\displaystyle \frac{1}{2\wp +1}}{\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1\right]\hfill \\ & ={\displaystyle \frac{1}{2\wp +1}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\left[{\displaystyle \frac{2\wp 1\otimes {\zeta }^{(2\wp +1)}0)+{\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1}{2\wp +2}}\right].\hfill \end{array}$$

If we multiply by ${\displaystyle \frac{1}{(2\wp )!}}$, we obtain

$$\begin{array}{cc}& {\displaystyle \frac{1}{(2\wp )!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\hfill \\ & \times {\int }_0^1{(\mathtt{1}-\mathtt{u})}^{2\wp }{\zeta }^{(2\wp +1)}((\mathtt{1}-\mathtt{u})1\otimes 0+\mathtt{u}\mathcal{U}\otimes 1)\mathtt{d}\mathtt{u}\hfill \\ & \le {\displaystyle \frac{1}{(2\wp +1)!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\left[{\displaystyle \frac{(2\wp +1)1\otimes {\zeta }^{(2\wp +1)}0)+{\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1}{2\wp +2}}\right],\hfill \end{array}$$

and this concludes the proof. □

Theorem 10. 

Assume that $\zeta$ is of class ${\mathcal{C}}^{2\wp +1}$ on the open interval $∆$ such that ${\zeta }^{2\wp +1}$ is a quasi-convex function on $∆$. Let $\mathcal{U}$ and $\mathcal{V}$ be self-adjoint operators with respective spectrums, $\mathcal{SP}\left(\mathcal{U}\right)\subset ∆$ and $\mathcal{SP}\left(\mathcal{V}\right)\subset ∆$, then one has

$$\begin{array}{cc}& \zeta \left(\mathcal{U}\right)\otimes 1-\sum _{\mathtt{k}=0}^{2\wp }{\displaystyle \frac{1}{\mathtt{k}!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{\mathtt{k}}\left(1\otimes {\zeta }^{\left(\mathtt{k}\right)}\left(0\right)\right)\hfill \\ & \le {\displaystyle \frac{1}{(2\wp +1)!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\hfill \\ & \times \left[{\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1+1\otimes {\zeta }^{(2\wp +1)}\left(0\right)+\left|{\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1-1\otimes {\zeta }^{(2\wp +1)}\left(0\right)\right|\right].\hfill \end{array}$$

Proof. 

Since ${\zeta }^{(2\wp +1)}$ is quasi-convex on ∆, then

$${\zeta }^{(2\wp +1)}((\mathtt{1}-\mathtt{u})0+\mathtt{ut})\le {\displaystyle \frac{1}{2}}\left({\zeta }^{(2\wp +1)}\left(\mathtt{t}\right)+{\zeta }^{(2\wp +1)}\left(0\right)+\left|{\zeta }^{(2\wp +1)}\left(\mathtt{t}\right)-{\zeta }^{(2\wp +1)}\left(0\right)\right|\right),$$

for all $\mathtt{t}\in ∆$ and $\mathtt{u}\in [0,1]$. If we multiply by ${(\mathtt{t}-0)}^{2\wp +1}$, we obtain

$$\begin{array}{cc}\hfill & {(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{(2\wp +1)}((\mathtt{1}-\mathtt{u})0+\mathtt{ut})\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{(\mathtt{t}-0)}^{2\wp +1}\left({\zeta }^{(2\wp +1)}\left(\mathtt{t}\right)+{\zeta }^{(2\wp +1)}\left(0\right)+\left|{\zeta }^{(2\wp +1)}\left(\mathtt{t}\right)-{\zeta }^{(2\wp +1)}\left(0\right)\right|\right),\hfill \end{array}$$

(13)

for all $\mathtt{t}\in ∆$ and $\mathtt{u}\in [0,1]$. By taking the integral ${\int }_{∆}{\int }_{∆}$ over ${\mathtt{dE}}_{\mathtt{t}}\otimes {\mathtt{dF}}_{\mathtt{s}}$ in (13), we obtain

$$\begin{array}{cc}\hfill & {\int }_{∆}{\int }_{∆}{(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{(2\wp +1)}((\mathtt{1}-\mathtt{u})0+\mathtt{ut})\mathtt{d}{\mathtt{E}}_{\mathtt{t}}\otimes \mathtt{d}{\mathtt{F}}_{\mathtt{s}}\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\int }_{∆}{\int }_{∆}{(\mathtt{t}-0)}^{2\wp +1}\left({\zeta }^{(2\wp +1)}\left(\mathtt{t}\right)+{\zeta }^{(2\wp +1)}\left(0\right)+\left|{\zeta }^{(2\wp +1)}\left(\mathtt{t}\right)-{\zeta }^{(2\wp +1)}\left(0\right)\right|\right)\hfill \\ \hfill & \times \mathtt{d}{\mathtt{E}}_{\mathtt{t}}\otimes \mathtt{d}{\mathtt{F}}_{\mathtt{s}}.\hfill \end{array}$$

(14)

Since, by Lemma 4,

$$\begin{array}{cc}& {\int }_{∆}{\int }_{∆}{(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{(2\wp +1)}((\mathtt{1}-\mathtt{u})0+\mathtt{ut})\mathtt{d}{\mathtt{E}}_{\mathtt{t}}\otimes \mathtt{d}{\mathtt{F}}_{\mathtt{s}}\hfill \\ & ={(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}{\zeta }^{(2\wp +1)}((\mathtt{1}-\mathtt{u})1\otimes 0+\mathtt{u}\mathcal{U}\otimes 1)\hfill \end{array}$$

and

$$\begin{array}{cc}& {\int }_{∆}{\int }_{∆}{(\mathtt{t}-0)}^{2\wp +1}\left({\zeta }^{(2\wp +1)}\left(\mathtt{t}\right)+{\zeta }^{(2\wp +1)}\left(0\right)+\left|{\zeta }^{(2\wp +1)}\left(\mathtt{t}\right)-{\zeta }^{(2\wp +1)}\left(0\right)\right|\right)\mathtt{d}{\mathtt{E}}_{\mathtt{t}}\otimes \mathtt{d}{\mathtt{F}}_{\mathtt{s}}\hfill \\ & ={\int }_{∆}{\int }_{∆}{(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{(2\wp +1)}\left(\mathtt{t}\right)\mathtt{d}{\mathtt{E}}_{\mathtt{t}}\otimes \mathtt{d}{\mathtt{F}}_{\mathtt{s}}+{\int }_{∆}{\int }_{∆}{(\mathtt{t}-0)}^{2\wp +1}{\zeta }^{(2\wp +1)}\left(0\right)\mathtt{d}{\mathtt{E}}_{\mathtt{t}}\otimes \mathtt{d}{\mathtt{F}}_{\mathtt{s}}\hfill \\ & +{\int }_{∆}{\int }_{∆}{(\mathtt{t}-0)}^{2\wp +1}\left|{\zeta }^{(2\wp +1)}\left(\mathtt{t}\right)-{\zeta }^{(2\wp +1)}\left(0\right)\right|\mathtt{d}{\mathtt{E}}_{\mathtt{t}}\otimes \mathtt{d}{\mathtt{F}}_{\mathtt{s}}\hfill \\ & ={(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}{\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1+{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\left(1\otimes {\zeta }^{(2\wp +1)}\left(0\right)\right)\hfill \\ & +{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\left|{\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1-1\otimes {\zeta }^{(2\wp +1)}\left(0\right)\right|\hfill \\ & ={(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\hfill \\ & \times \left[{\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1+1\otimes {\zeta }^{(2\wp +1)}\left(0\right)+\left|{\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1-1\otimes {\zeta }^{(2\wp +1)}\left(0\right)\right|\right],\hfill \end{array}$$

then by (14), we obtain

$$\begin{array}{cc}& {(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}{\zeta }^{(2\wp +1)}((\mathtt{1}-\mathtt{u})1\otimes 0+\mathtt{u}\mathcal{U}\otimes 1)\hfill \\ & \le {(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\hfill \\ & \times \left[{\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1+1\otimes {\zeta }^{(2\wp +1)}\left(0\right)+\left|{\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1-1\otimes {\zeta }^{(2\wp +1)}\left(0\right)\right|\right],\hfill \end{array}$$

for $\mathtt{u}\in [0,1]$. If we multiply the above inequality by ${(\mathtt{1}-\mathtt{u})}^{2\wp }$ and integrate, then we obtain

$$\begin{array}{cc}& {(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}{\int }_0^1{(\mathtt{1}-\mathtt{u})}^{2\wp }{\zeta }^{(2\wp +1)}((\mathtt{1}-\mathtt{u})1\otimes 0+\mathtt{u}\mathcal{U}\otimes 1)\mathtt{d}\mathtt{u}\hfill \\ & \le {\displaystyle \frac{1}{2\wp +1}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp +1}\hfill \\ & \times \left[{\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1+1\otimes {\zeta }^{(2\wp +1)}\left(0\right)+\left|{\zeta }^{(2\wp +1)}\left(\mathcal{U}\right)\otimes 1-1\otimes {\zeta }^{(2\wp +1)}\left(0\right)\right|\right],\hfill \end{array}$$

and the proof is now complete. □

Example 1. 

Under the identical hypothesis of Theorem 8, and if $\zeta (\mu )=exp(\mu )$, then by (6), we obtain

$$\begin{array}{cc}\hfill exp\left(\mathcal{U}\right)\otimes 1& =\sum _{\mathtt{k}=\mathtt{0}}^{2\wp -1}{\displaystyle \frac{1}{\mathtt{k}!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{\mathtt{k}}(1\otimes exp\left(0\right))\hfill \\ \hfill & +{\displaystyle \frac{1}{(2\wp -1)!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp }\hfill \\ \hfill & \times {\int }_0^1{(1-\mathtt{u})}^{2\wp -1}exp((1-\mathtt{u})1\otimes 0+\mathtt{u}\mathcal{U}\otimes 1)\mathtt{d}\mathtt{u},\hfill \end{array}$$

(15)

for any self-adjoint operators $\mathcal{U},\mathcal{V}$. If ${\alpha }_{2\wp +1}\le \mathcal{U},\mathcal{V}\le {\beta }_{2\wp +1}$, then by (5), we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\left(2\wp \right)!}}exp\left({\alpha }_{2\wp +1}\right){(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp }\hfill \\ \hfill & \le exp\left(\mathcal{U}\right)\otimes 1-\sum _{\mathtt{k}=\mathtt{0}}^{2\wp -1}{\displaystyle \frac{1}{\mathtt{k}!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{\mathtt{k}}(1\otimes exp\left(0\right))\hfill \\ \hfill & \le {\displaystyle \frac{1}{\left(2\wp \right)!}}exp\left({\beta }_{2\wp +1}\right){(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp }\hfill \end{array}$$

(16)

From (9), we obtain

$$\begin{array}{cc}\hfill & exp\left(\mathcal{U}\right)\otimes 1-\sum _{\mathtt{k}=\mathtt{0}}^{2\wp -1}{\displaystyle \frac{1}{\mathtt{k}!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{\mathtt{k}}(1\otimes exp\left(0\right))\hfill \\ \hfill & \le {\displaystyle \frac{1}{\left(2\wp \right)!}}{(\mathcal{U}\otimes 1-1\otimes 0)}^{2\wp }\left[{\displaystyle \frac{2\wp (1\otimes exp(0\left)\right)+exp\left(\mathcal{U}\right)\otimes 1}{2\wp +1}}\right].\hfill \end{array}$$

(17)

for any self-adjoint operators $\mathcal{U},\mathcal{V}$. If $w,t>0$ and if we take in (15), $\mathcal{U}=lnw,\mathcal{U}=lnt$, then we have

$$\begin{array}{cc}\hfill w\otimes 1& =\sum _{\mathtt{k}=\mathtt{0}}^{2\wp -1}{\displaystyle \frac{1}{\mathtt{k}!}}{(lnw\otimes 1-1\otimes lnt)}^{\mathtt{k}}(1\otimes t)\hfill \\ & +{\displaystyle \frac{1}{(2\wp -1)!}}{(lnw\otimes 1-1\otimes lnt)}^{2\wp }\hfill \\ & \times {\int }_0^1{(1-\mathtt{u})}^{2\wp -1}exp((1-\mathtt{u})1\otimes lnt+\mathtt{u}lnw\otimes 1)\mathtt{d}\mathtt{u}.\hfill \end{array}$$

and for $\wp =1$, we have

$$\begin{array}{cc}\hfill w\otimes 1& =(1\otimes t)+(lnw\otimes 1-1\otimes lnt)(1\otimes t)\hfill \\ & +{(lnw\otimes 1-1\otimes lnt)}^2\hfill \\ & \times {\int }_0^1(1-\mathtt{u})exp((1-\mathtt{u})1\otimes lnt+\mathtt{u}lnw\otimes 1)\mathtt{d}\mathtt{u}.\hfill \end{array}$$

If $0<\psi \le w,t\le \Psi$, then by (16), we obtain

$$\begin{array}{cc}& {\displaystyle \frac{1}{\left(2\wp \right)!}}\psi {(lnw\otimes 1-1\otimes lnt)}^{2\wp }\hfill \\ & \le w\otimes 1-\sum _{\mathtt{k}=\mathtt{0}}^{2\wp -1}{\displaystyle \frac{1}{\mathtt{k}!}}{(lnw\otimes 1-1\otimes lnt)}^{\mathtt{k}}(1\otimes t)\hfill \\ & \le {\displaystyle \frac{1}{\left(2\wp \right)!}}\Psi {(lnw\otimes 1-1\otimes lnt)}^{2\wp }\hfill \end{array}$$

and for $\wp =1$, we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}\psi {(lnw\otimes 1-1\otimes lnt)}^2\le w\otimes 1-1\otimes t-(lnw\otimes 1-1\otimes lnt)(1\otimes t)\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}\Psi {(lnw\otimes 1-1\otimes lnt)}^2.\hfill \end{array}$$

(18)

From (17), we obtain

$$\begin{array}{cc}& w\otimes 1-\sum _{\mathtt{k}=\mathtt{0}}^{2\wp -1}{\displaystyle \frac{1}{\mathtt{k}!}}{(lnw\otimes 1-1\otimes lnt)}^{\mathtt{k}}(1\otimes t)\hfill \\ & \le {\displaystyle \frac{1}{\left(2\wp \right)!}}{(lnw\otimes 1-1\otimes lnt)}^{2\wp }\left[{\displaystyle \frac{2\wp (1\otimes t)+w\otimes 1}{2\wp +1}}\right]\hfill \end{array}$$

and for $\wp =1$,

$$\begin{array}{cc}& w\otimes 1-1\otimes t-(lnw\otimes 1-1\otimes lnt)(1\otimes t)\hfill \\ & \le {\displaystyle \frac{1}{2}}{(lnw\otimes 1-1\otimes lnt)}^2\left[{\displaystyle \frac{2(1\otimes t)+w\otimes 1}{3}}\right],\hfill \end{array}$$

for all $w,t>0$. Using the commutativity of $w\otimes 1$ and $1\otimes t$, we may deduce from (18) that

$$\begin{array}{cc}& {\displaystyle \frac{1}{2}}\psi {(lnw\otimes 1-1\otimes lnt)}^2\left(1\otimes t^{-1}\right)\hfill \\ & \le 1\otimes lnt-1-lnw\otimes 1+w\otimes t^{-1}\hfill \\ & \le {\displaystyle \frac{1}{2}}\Psi {(lnw\otimes 1-1\otimes lnt)}^2\left(1\otimes t^{-1}\right),\hfill \end{array}$$

provided that $0<\psi \le w,t\le \Psi$.

4. Preliminary Results

In this section, we will recall some fundamental definitions and conclusions related to mixed norm function spaces with varying exponents. Reference [79] contains the majority of the properties linked to function spaces, whereas mixed Lebesgue spaces are covered in the following articles [39,80,81].

4.1. Semi-Modular Spaces

The variable Lebesgue spaces are categorized as semi-modular spaces, which is a generalization of a normed space. For this purpose, we first review certain modular space definitions and results.

Definition 6. 

Let $\mathcal{M}$ be a vector space over some field $\mathcal{K}$ (real or complex). The mapping $\rho :\mathcal{M}⟶[0,\infty ]$ is said to be semi-modular on $\mathcal{M}$ if

• $\rho \left(0\right)=0$;
• $\rho (\alpha \mu )=\rho (\mu )$ for all $\mu \in \mathcal{M}$ and $\alpha \in \mathcal{K},|\alpha |=1$;
• if $\rho (\alpha \mu )=0$ for all $\alpha >0$, then $\mu =0$;
• function $[0,\infty )\ni \alpha \mapsto \rho (\alpha \mu )$ is left-continuous for every $\mu \in \mathcal{M}$;
• function $[0,\infty )\ni \alpha \mapsto \rho (\alpha \mu )$ is non-decreasing for every $\mu \in \mathcal{M}$.

If a vector space $\mathcal{M}$ has $\rho$ as a semi-modular on it, then the space

$${\mathcal{M}}_{\rho }:=\left\{\mu \in \mathcal{M}:{\exists }_{\alpha >0}\rho (\alpha \mu )<\infty \right\},$$

known as a semi-modular space, is connected to the functional ${\parallel ·\parallel }_{\rho }:{\mathcal{K}}_{\rho }\to [0,\infty ]$ and is defined as follows:

$${\parallel \mathtt{s}\parallel }_{\rho }:=inf\{\alpha >0:\rho (\mathtt{s}/\alpha )\le 1\}\mathrm{for}\mathtt{s}\in {\mathcal{K}}_{\rho }.$$

Proposition 1. 

Suppose that on vector space $\mathcal{M}$, $\rho$ is semi-modular. Consequently, for each $\mu \in \mathcal{M}$, we have

$$\rho (\mu )\le 1⟺{\parallel \mu \parallel }_{\rho }\le 1.$$

Proof. 

Assume that $\rho (\mu )\le 1$. Then, ${\parallel \mu \parallel }_{\rho }\le 1$ is inferred from the definition of the norm ${\parallel ·\parallel }_{\rho }$. Conversely, if we assume that ${\parallel \mu \parallel }_{\rho }\le 1$, then for each $\alpha >1$ and due to the left continuity of the function, we have

$$\rho \left(\mu /\alpha \right)\le 1.$$

□

4.2. Variable Exponent Spaces

Now let us review the idea of ${\mathtt{L}}^{\mathtt{p}(·)}$ space. To do this, we take into consideration that $∆\subset \Omega$ is a measurable set and we define the measurable function $\mathtt{p}:∆⟶(0,\infty ]$, which is normally described as follows:

$${\mathtt{p}}_{+}=\underset{\kappa \in ∆}{esssup}\mathtt{p}(\kappa ),{\mathtt{p}}_{-}=\underset{\kappa \in ∆}{essinf}\mathtt{p}(\kappa ).$$

We refer to any Lebesgue measurable functions $\mathtt{p}:∆⟶(0,\infty )]$ as variable exponents such that ${\mathtt{p}}_{-}>0$. For the set ∆, ${\mathcal{P}}_0(∆)$ represents the space of all varying exponents functions. The notation $\mathcal{P}(∆)$ marks the pack of all varying exponent functions $\mathtt{p}$ such that ${\mathtt{p}}_{-}\ge 1$.

Consider the vector space ${\mathtt{L}}^0(∆)$, which contains all measurable functions over ∆. For $\mathtt{p}\in {\mathcal{P}}_0(∆)$, over the space ${\mathtt{L}}^0(∆),$ we define the semi-modular as follows:

$${\rho }_{\mathtt{p}(·)}(\zeta ):={\int }_{∆}{\Psi }_{\mathtt{p}(\mu )}\left(\right|\zeta (\mu )\left|\right)\mathtt{d}\mu$$

where

$${\Psi }_{\mathtt{p}}(\sigma ):=\left\{\begin{array}{cc}{\sigma }^{\mathtt{p}},\hfill & \mathrm{if}0<\mathtt{p}<\infty ,\hfill \\ 0,\hfill & \mathrm{if}\sigma \le 1,\mathtt{p}=\infty ,\hfill \\ \infty ,\hfill & \mathrm{if}\sigma >1,\mathtt{p}=\infty \hfill \end{array}\right.$$

for each $\sigma \ge 0$. This modular space is described by the semi-modular ${\rho }_{\mathtt{p}(·)}$, which in turn defines the ${\mathtt{L}}^{\mathtt{p}(·)}(∆)$ space as follows:

$$\zeta \in {\mathtt{L}}^{\mathtt{p}(·)}(∆)⟺{\exists }_{\alpha >0}{\rho }_{\mathtt{p}(·)}(\alpha \zeta )={\int }_{∆}{\phi }_{\mathtt{p}(\mu )}(\alpha |\zeta (\mu )\left|\right)\mathtt{d}\mu <\infty .$$

The following Luxemburg quasi-norm makes the ${\mathtt{L}}^{\mathtt{p}(·)}(∆)$ space a quasi-Banach space:

$${\parallel \zeta \parallel }_{{\mathtt{L}}^{\mathtt{p}(·)}(∆)}:=inf\left\{\alpha >0:{\rho }_{\mathtt{p}(·)}(\zeta /\alpha )\le 1\right\}\mathrm{for}\zeta \in {\mathtt{L}}^{\mathtt{p}(·)}(∆).$$

It is the particular case of the Musielak–Orlicz space, and whenever $\mathtt{p}\in \mathcal{P}(∆)$, then ${\mathtt{L}}^{\mathtt{p}(·)}(∆)$ becomes Banach space. When the exponent is constant, it reduces to the classical Lebesgue space.

Notably, the definition of ${\mathtt{L}}^{\mathtt{p}(·)}(∆)$ space becomes simpler if ${\mathtt{p}}_{+}<\infty$ so that $\zeta \in {\mathtt{L}}^{\mathtt{p}(·)}(∆)$ if, and only if

$${\int }_{∆}{|\zeta (\mu )|}^{\mathtt{p}(\mu )}\mathtt{d}\mu <\infty .$$

Certain presumptions on the exponent function $\mathtt{p}$ are required for a few claims in this work. Specifically, function $\mathtt{p}:∆⟶\mathtt{R}$ is locally log-Hölder continuous on ∆, if there exists $c_{log}\left(\mathtt{p}\right)>0$ and for all $\mu ,\nu \in ∆$, we have

$$|\mathtt{p}(\mu )-\mathtt{p}(\nu )|\le {\displaystyle \frac{c_{log}\left(\mathtt{p}\right)}{log(\mathrm{e}+1/|\mu -\nu \left|\right)}}.$$

We say that $\mathtt{p}$ satisfies the log-Hölder continuous at infinity (or has a log decay at infinity), if there exists ${\mathtt{p}}_{\infty }\in ∆$ and a constant $c_{log}>0$ and for all $\nu \in ∆$. We have

$$\left|\mathtt{p}(\mu )-{\mathtt{p}}_{\infty }\right|\le {\displaystyle \frac{c_{log}\left(\mathtt{p}\right)}{log(\mathrm{e}+|\mu \left|\right)}}.$$

If both of the aforementioned criteria are met by the function $\mathtt{p}$, we declare that $\mathtt{p}$ is globally log-Hölder continuous ${\mathcal{C}}_{log}\left(\mathtt{p}\right)$ on ∆. The corresponding class of functions is defined as follows:

$${\mathcal{P}}_0^{log}(∆):=\left\{\mathtt{p}\in {\mathcal{P}}_0(∆):{\displaystyle \frac{1}{\mathtt{p}}}\in {\mathcal{C}}_{log}\left(\mathtt{p}\right)\right\}.$$

For the semi-modular ${\rho }_{\mathtt{p}(·)}$ and the quasi-norm ${\parallel ·\parallel }_{{\mathtt{L}}^{\mathtt{p}(·)}(∆)}$, we now formulate several well-known and practical statements.

Proposition 2. 

Let $\mathtt{p}\in {\mathcal{P}}_0(\Omega )$ with ${\mathtt{p}}_{+}<\infty$ and let $\zeta \in {\mathtt{L}}^{\mathtt{p}(·)}(\Omega )$. Then,

$$min\left\{{\left({\rho }_{\mathtt{p}(·)}(\zeta )\right)}^{{\displaystyle \frac{1}{{\mathtt{p}}_{-}}}},{\left({\rho }_{\mathtt{p}(·)}(\zeta )\right)}^{{\displaystyle \frac{1}{{\mathtt{p}}_{+}}}}\right\}\le {\parallel \zeta \parallel }_{{\mathtt{L}}^{\mathtt{p}(·)}(\Omega )}\le max\left\{{\left({\rho }_{\mathtt{p}(·)}(\zeta )\right)}^{{\displaystyle \frac{1}{{\mathtt{p}}_{-}}}},{\left({\rho }_{\mathtt{p}(·)}(\zeta )\right)}^{{\displaystyle \frac{1}{{\mathtt{p}}_{+}}}}\right\}.$$

We now recall a generalized Orlicz–Zygmund space with variable integrability.

Definition 7 

([82]). The space ${\mathtt{L}}^{\mathtt{p}(\mu )}{log}^{\beta }\mathtt{L}$ is defined for each $\beta \in \mathtt{R}$ as

$${\mathtt{L}}^{\mathtt{p}(\mu )}{log}^{\beta }\mathtt{L}\left(∆,\Omega \right):=\left\{\nu :∆\to \Omega :\nu \mathit{is}\mathit{measurable}\mathit{and}{\int }_{∆}{|\nu |}^{\mathtt{p}(\mu )}{log}^{\beta }(e+|\nu \left|\right)\mathtt{d}\nu <\infty \right\}$$

and the associated Luxemburg norm is defined as follows:

$${\parallel \nu \parallel }_{{\mathtt{L}}^{\mathtt{p}(\mu )}{log}^{\beta }\mathtt{L}\left(∆,\Omega \right)}:=inf\left\{\mathsf{\lambda }>0:{\int }_{∆}{\left|{\displaystyle \frac{\nu }{\mathsf{\lambda }}}\right|}^{\mathtt{p}(\mu )}{log}^{\beta }\left(e+{\displaystyle \frac{|\nu |}{\mathsf{\lambda }}}\right)\mathtt{d}\mu ⩽1\right\},$$

it becomes a Banach space.

4.3. Mixed Norm Function Spaces

Definition 8 

([81]). 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 ${\mathtt{L}}^{\mathtt{p}(·)}$ as follows:

$${\rho }_{\ell \mathtt{q}(·)\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\left(\left({\zeta }_{\mathtt{j}}\right)\right):=\sum _{\mathtt{j}=\mathtt{1}}^{\infty }inf\left\{{\alpha }_{\mathtt{j}}>0:{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\zeta }_{\mathtt{j}}}{{\alpha }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\},$$

where the corresponding Luxemburg norm, typically defined as

$${∥\left({\zeta }_{\mathtt{j}}\right)∥}_{{\ell }^{\mathtt{q}(·)}\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}:=inf\left\{\nu >0:{\rho }_{\ell \mathtt{q}(·)}\left({\mathtt{L}}^{\mathtt{p}(·)}\right)\left(\left({\zeta }_{\mathtt{j}}\right)/\nu \right)\le 1\right\},\left({\zeta }_{\mathtt{j}}\right)\in {\ell }^{\mathtt{q}(·)}\left({\mathtt{L}}^{\mathtt{p}(·)}\right).$$

Under certain assumptions [39], the space ${\ell }^{\mathtt{q}(·)}\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$ becomes a normed space.

Definition 9 

([81]). Consider $\mathtt{p},\mathtt{q},\mathtt{u}\in \mathcal{P}\left({\mathsf{R}}^{\mathtt{n}}\right)$ where $\mathtt{p}(\mu )\le \mathtt{u}(\mu )$. The mixed norm Morrey-sequence space ${\ell }_{\mathtt{q}(·)}\left({\mathbf{M}}_{\mathtt{p}(·),\mathtt{u}(·)}\right)$ encompasses all measurable sequences ${\left({\zeta }_{\sigma }\right)}_{\kappa }\subset \mathcal{K}\left({\mathsf{R}}^{\mathtt{n}}\right)$ such that

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

5. Main Results Related to Mixed-Orlicz–Zygmund $\left({\ell }_{\mathtt{q}(·)}{\mathbf{log}}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)\right)$ Space

The aim of this section is to develop a new norm structure by merging two different types of function spaces and analyze whether this space is complete, separable, and compact. First, taking into account Definition 7 and Definition 8, we propose a new type of function space and its related modular and norm, which we use in our major results.

Definition 10. 

Let $\mathtt{p},\mathtt{q}\in \mathcal{P}\left({\mathsf{R}}^{\mathtt{n}}\right)$. The mixed generalized Orlic Zygmund sequence space ${\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$ is defined on the sequences $\left({\zeta }_{\mathtt{j}}\right)$ of ${\mathtt{L}}^{\mathtt{p}(·)}$ as follows:

$${\rho }_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\left(\left({\zeta }_{\mathtt{j}}\right)\right):=\sum _{\mathtt{j}=\mathtt{1}}^{\infty }inf\left\{{\alpha }_{\mathtt{j}}>0:{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\zeta }_{\mathtt{j}}}{{\alpha }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }^{\mathtt{j}}|}{{\alpha }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}.$$

If we consider the following convention ${\alpha }^{{\displaystyle \frac{1}{\infty }}}=1$, the norm is defined as follows:

$${∥{\left({\zeta }_{\mathtt{j}}\right)}_{\mathtt{j}}∥}_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}:=inf\left\{\nu >0|{\rho }_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\left({\displaystyle \frac{1}{\nu }}{\left({\zeta }_{\mathtt{j}}\right)}_{\mathtt{j}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }^{\mathtt{j}}|}{\nu }}\right)\le 1\right\}.$$

If ${\mathtt{q}}^{+}<\infty$, then

$$inf\left\{\alpha >0\left|{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{\zeta }{{\alpha }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|\zeta |}{{\alpha }^{\frac{1}{\mathtt{q}(·)}}}}\right)⩽1\right.\right\}={∥{|\zeta |}^{\mathtt{q}(·)}∥}_{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}.$$

5.1. Completeness

First, we demonstrate that this modular space is a quasi-Banach space.

Theorem 11. 

If $\mathtt{p},\mathtt{q}\in {\mathcal{P}}_0$, then ${\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$ is a quasi-Banach space.

Proof. 

To demonstrate ${\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$ is a quasi-Banach space. We first show that it is a quasi-normed space. By Definition 6, we just need to consider the quasi-convexity. Let $\kappa \in \left(0,min\left\{{\displaystyle \frac{{\mathtt{p}}^{-}}{2}},{\displaystyle \frac{{\mathtt{q}}^{-}}{2}},1\right\}\right]$ and define $\tilde{\mathtt{p}}={\displaystyle \frac{\mathtt{p}}{\kappa }}$ and $\tilde{\mathtt{q}}={\displaystyle \frac{\mathtt{q}}{\kappa }}$. Then, clearly ${\displaystyle \frac{1}{\widetilde{\mathtt{p}(·)}}}+{\displaystyle \frac{1}{\widetilde{\mathtt{q}(·)}}}⩽1$. Thus, we have

$$\begin{array}{cc}& {∥{\left({\eta }_{\nu }\right)}_{\nu }+{\left({\zeta }_{\nu }\right)}_{\nu }∥}_{{\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}={∥{\left|{\left({\eta }_{\nu }\right)}_{\nu }+{\left({\zeta }_{\nu }\right)}_{\nu }\right|}^{\kappa }∥}_{{\ell }^{\tilde{\mathtt{q}}(·)}{log}^{\beta }\left({\mathtt{L}}^{(\tilde{\mathtt{p}}(·)}\right)}^{{\displaystyle \frac{1}{\kappa }}}\hfill \\ & ⩽{∥{\left({\left|{\eta }_{\nu }\right|}^{\kappa }\right)}_{\nu }+{\left({\left|{\zeta }_{\nu }\right|}^{\kappa }\right)}_{\nu }∥}_{{\ell }^{\tilde{\mathtt{q}}(·)}{log}^{\beta }\left({\mathtt{L}}^{(\tilde{\mathtt{p}}(·)}\right)}^{{\displaystyle \frac{1}{\kappa }}}\hfill \\ & ⩽{\left({∥{\left({\left|{\eta }_{\nu }\right|}^{\kappa }\right)}_{\nu }∥}_{{\ell }^{\tilde{\mathtt{q}}(·)}{log}^{\beta }\left({\mathtt{L}}^{\tilde{\mathtt{p}}(·)}\right)}+{∥{\left({\left|{\zeta }_{\nu }\right|}^{\kappa }\right)}_{\nu }∥}_{{\ell }^{\tilde{\mathtt{q}}(·)}{log}^{\beta }\left({\mathtt{L}}^{\tilde{\mathtt{p}}(·)}\right)}\right)}^{{\displaystyle \frac{1}{\kappa }}}\hfill \\ & ={\left({∥{\left({\eta }_{\nu }\right)}_{\nu }∥}_{{\ell }^{\tilde{\mathtt{q}}(·)}{log}^{\beta }\left({\mathtt{L}}^{(\tilde{\mathtt{p}}(·)}\right)}^{\kappa }+{∥{\left({\zeta }_{\nu }\right)}_{\nu }∥}_{{\ell }^{\tilde{\mathtt{q}}(·)}{log}^{\beta }\left({\mathtt{L}}^{(\tilde{\mathtt{p}}(·)}\right)}^{\kappa }\right)}^{{\displaystyle \frac{1}{\kappa }}}\hfill \\ & ⩽2^{{\displaystyle \frac{1-\kappa }{\kappa }}}\left({∥{\left({\eta }_{\nu }\right)}_{\nu }∥}_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}+{∥{\left({\zeta }_{\nu }\right)}_{\nu }∥}_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\right),\hfill \end{array}$$

and this completes the proof. The next step is to demonstrate the space’s completeness. For this purpose, we fix a Cauchy sequence ${\zeta }^{\mathtt{i}}$ in $\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$. Hence, for any $\epsilon >0$, there exists ℘ such that for all $\mathtt{i},\mathtt{j}\ge \wp$, we have

$${∥{\zeta }^{\mathtt{i}}-{\zeta }^{\mathtt{j}}∥}_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}<\epsilon$$

By the quasi-norm’s definition

$${\rho }_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\left({\displaystyle \frac{{\zeta }^{\mathtt{i}}-{\zeta }^{\mathtt{j}}}{\epsilon }}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }^{\mathtt{i}}-{\zeta }^{\mathtt{j}}|}{\epsilon }}\right)\le 1,\mathrm{for}\mathtt{i},\mathtt{j}\ge \wp ,$$

and hence

$$\sum _{\mu =1}^{\infty }inf\left\{{\sigma }_{\mu }:{\rho }_{{\mathtt{L}}^{\mathtt{p}(·)}}\left({\displaystyle \frac{{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{\mathtt{j}}}{\epsilon {\sigma }_{\mu }^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{\mathtt{j}}|}{\epsilon {\sigma }_{\mu }^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\le 1.$$

In particular, for each $\mathsf{\delta }>1$, we have

$${\rho }_{{\mathtt{L}}^{\mathtt{p}(·)}}\left({\displaystyle \frac{{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{\mathtt{j}}}{\epsilon {\mathsf{\delta }}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{\mathtt{j}}|}{\epsilon {\mathsf{\delta }}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1.$$

Hence, by the definition of the space ${\mathtt{L}}^{\mathtt{p}(·)}$, we obtain

$${\displaystyle \frac{1}{{\mathsf{\delta }}^{\frac{1}{{\mathtt{q}}^{-}}}}}{∥{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{\mathtt{j}}∥}_{{\mathtt{L}}^{\mathtt{p}(·)}}\le {∥{\displaystyle \frac{{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{\mathtt{j}}}{{\mathsf{\delta }}^{\frac{1}{\mathtt{q}(·)}}}}∥}_{{\mathtt{L}}^{\mathtt{p}(·)}}\le \epsilon .$$

Thus, passing with $\mathsf{\delta }\to 1$, we obtain ${∥{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{\mathtt{j}}∥}_{{\mathtt{L}}^{\mathtt{p}(·)}}\le \epsilon$. Thus, we demonstrate that for each $\mu ,{\zeta }_{\mu }^{\mathtt{i}}$ is a Cauchy sequence in ${\mathtt{L}}^{\mathtt{p}(·)}$. Thus, for every $\mu$, there exists ${\zeta }_{\mu }\in {\mathtt{L}}^{\mathtt{p}(·)}$ such that

$${\zeta }_{\mu }^{\mathtt{i}}\to {\zeta }_{\mu }\mathrm{in}{\mathtt{L}}^{\mathtt{p}(·)}.$$

Now, we shall prove that

$$\begin{array}{c}\hfill {\zeta }^{\mathtt{i}}\to \zeta \mathrm{in}{\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right),\end{array}$$

(19)

where $\zeta ={\left({\zeta }_{\mu }\right)}_{\mu }$. For this purpose, we require the following:

$$\begin{array}{cc}\hfill & inf\left\{{\sigma }_{\mu }:{\rho }_{{\mathtt{L}}^{\mathtt{p}(·)}}\left({\displaystyle \frac{{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }}{\epsilon {\sigma }_{\mu }^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{\mathtt{j}}|}{\epsilon {\sigma }_{\mu }^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\hfill \\ & \le \underset{\mathtt{j}\to \infty }{lim\; inf}inf\left\{{\sigma }_{\mu }\left(\mathtt{j}\right):{\rho }_{{\mathtt{L}}^{\mathtt{p}(·)}}\left({\displaystyle \frac{{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{\mathtt{j}}}{\epsilon {\sigma }_{\mu }{\left(\mathtt{j}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{\mathtt{j}}|}{\epsilon {\sigma }_{\mu }{\left(\mathtt{j}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}.\hfill \end{array}$$

(20)

In order to prove (20), we introduce the following notation:

$$\mathcal{A}=\underset{\mathtt{j}\to \infty }{lim\; inf}inf\left\{{\sigma }_{\mu }\left(\mathtt{j}\right):{\rho }_{{\mathtt{L}}^{\mathtt{p}(·)}}\left({\displaystyle \frac{{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{\mathtt{j}}}{\epsilon {\sigma }_{\mu }{\left(\mathtt{j}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{\mathtt{j}}|}{\epsilon {\sigma }_{\mu }{\left(\mathtt{j}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}.$$

Let ${\mathtt{j}}_{\mathtt{k}}$ be a subsequence such that

$$\mathcal{A}=\underset{\mathtt{k}\to \infty }{lim}inf\left\{{\sigma }_{\mu }\left({\mathtt{j}}_{\mathtt{k}}\right):{\rho }_{{\mathtt{L}}^{\mathtt{p}(·)}}\left({\displaystyle \frac{{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{{\mathtt{j}}_{\mathtt{k}}}}{\epsilon {\sigma }_{\mu }{\left({\mathtt{j}}_{\mathtt{k}}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{\mathtt{j}}|}{\epsilon {\sigma }_{\mu }{\left({\mathtt{j}}_{\mathtt{k}}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}.$$

Hence, for each $\upsilon >0$ there exists $\mathtt{N}$ such that for all $\mathtt{k}\ge \mathtt{N}$, one has

$$\left\{{\sigma }_{\mu }\left({\mathtt{j}}_{\mathtt{k}}\right):{\rho }_{{\mathtt{L}}^{\mathtt{p}(·)}}\left({\displaystyle \frac{{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{{\mathtt{j}}_{\mathtt{k}}}}{\epsilon {\sigma }_{\mu }{\left({\mathtt{j}}_{\mathtt{k}}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{\mathtt{j}}|}{\epsilon {\sigma }_{\mu }{\left({\mathtt{j}}_{\mathtt{k}}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}<\mathcal{A}+\upsilon .$$

Thus,

$${\rho }_{{\mathtt{L}}^{\mathtt{p}(·)}}\left({\displaystyle \frac{{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{{\mathtt{j}}_{\mathtt{k}}}}{\epsilon {(\mathcal{A}+\upsilon )}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{\mathtt{j}}|}{\epsilon {(\mathcal{A}+\upsilon )}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1.$$

Next, we apply the Fatou Lemma and the fundamental characteristics of the ${\mathtt{L}}^{\infty }$ norm, and we obtain

$${\rho }_{{\mathtt{L}}^{\mathtt{p}(·)}}\left({\displaystyle \frac{{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }}{\epsilon {(\mathcal{A}+\upsilon )}^{\frac{1}{{\mathtt{q}}^{(·)}}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }|}{\epsilon {(\mathcal{A}+\upsilon )}^{\frac{1}{{\mathtt{q}}^{(·)}}}}}\right)\le 1.$$

Hence,

$$inf\left\{{\sigma }_{\mu }:{\rho }_{{\mathtt{L}}^{\mathtt{p}(·)}}\left({\displaystyle \frac{{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }}{\epsilon {\sigma }_{\mu }^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }|}{\epsilon {\sigma }_{\mu }^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\le \mathcal{A}+\upsilon .$$

Finally, applying $\upsilon \to 0$, we obtain (20). Now, we can prove (19). Since ${\zeta }^{\mathtt{i}}$ is a Cauchy sequence in ${\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$, for $\mathtt{i},\mathtt{j}\ge \wp$ we have

$$\sum _{\mu =1}^{\infty }inf\left\{{\sigma }_{\mu }:{\rho }_{{\mathtt{L}}^{\mathtt{p}(·)}}\left({\displaystyle \frac{{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{\mathtt{j}}}{\epsilon {\sigma }_{\mu }^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }^{\mathtt{j}}|}{\epsilon {\sigma }_{\mu }^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\le 1.$$

Therefore, by (20), for each $\mathtt{N}$ and $\mathtt{i}\ge \wp$, we have

$$\sum _{\mu =1}^{\mathtt{N}}inf\left\{{\sigma }_{\mu }:{\rho }_{{\mathtt{L}}^{\mathtt{p}(·)}}\left({\displaystyle \frac{{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }}{\epsilon {\sigma }_{\mu }^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mu }^{\mathtt{i}}-{\zeta }_{\mu }|}{\epsilon {\sigma }_{\mu }^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\le 1.$$

Then, passing $\mathtt{N}\to \infty$, we obtain

$${∥{\zeta }^{\mathtt{i}}-\zeta ∥}_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\le \epsilon \mathrm{for}\mathtt{i}\ge \wp .$$

□

5.2. Separability

The next result shows that the mixed norm space ${\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$ is separable when the exponents satisfy ${\mathtt{p}}^{+},{\mathtt{q}}^{+}<\infty$.

Theorem 12. 

If $\mathtt{p},\mathtt{q}\in {\mathcal{P}}_0$ and ${\mathtt{p}}^{+},{\mathtt{q}}^{+}<\infty$, then ${\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$ is separable.

Proof. 

For each $\mathtt{k}\in \mathtt{N}$, we define

$${\mathcal{V}}_{\mathtt{k}}=\left\{\zeta \in {\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right);{\zeta }_{\mathtt{i}}=0,\mathtt{i}>\mathtt{k}\right\}.$$

It is clear that ${\mathcal{V}}_{\mathtt{k}}$ represents a closed subset of ${\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$ and

$$\underset{\mathtt{k}\mathrm{times}}{\underbrace{{\mathtt{L}}^{\mathtt{p}(·)}\times {\mathtt{L}}^{\mathtt{p}(·)}\times \dots \times {\mathtt{L}}^{\mathtt{p}(·)}}}\times \left\{0\right\}\times \dots \times \left\{0\right\}\times \dots \subset {\mathcal{V}}_{\mathtt{k}}.$$

We divide the proof into two parts. First, we show that for each $\mathtt{k}\in \mathtt{N}$, the space ${\mathcal{V}}_{\mathtt{k}}$ is separable. Since ${\mathtt{p}}^{+}<\infty$, we know that ${\mathtt{L}}^{\mathtt{p}(·)}$ is separable. So there is a countable subset of ${\mathtt{L}}^{\mathtt{p}(·)}$, say, $\mathcal{K}$ such that which ${\overline{\mathcal{K}}}^{{\parallel ·\parallel }_{{\mathtt{L}}^{\mathtt{p}(·)}}}={\mathtt{L}}^{\mathtt{p}(·)}$. Set

$${\mathcal{K}}_{\mathtt{k}}:=\underset{\mathtt{k}\mathrm{times}}{\underbrace{\mathcal{K}\times \mathcal{K}\times \dots \times \mathcal{K}}}\times \left\{0\right\}\times \dots \times \left\{0\right\}\times \dots \subset {\mathcal{V}}_{\mathtt{k}}.$$

Obviously ${\mathcal{K}}_{\mathtt{k}}$ is countable, so it is enough to show that ${\overline{\mathcal{K}}}^{{\parallel ·\parallel }_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}}={\mathcal{V}}_{\mathtt{k}}$. Let us fix $\epsilon >0$ and $\zeta \in {\mathcal{V}}_{\mathtt{k}}$. Then, there exists $\eta \in {\mathcal{K}}_{\mathtt{k}}$ such that for $\nu \in \{1,2,\dots ,\mathtt{k}\}$, we have

$${∥{\zeta }_{\nu }-{\eta }_{\nu }∥}_{{\mathtt{L}}^{\mathtt{p}(·)}}\le {\displaystyle \frac{\epsilon }{{\mathtt{k}}^{\frac{1}{{\mathtt{q}}^{-}}}}}.$$

Then, we have ${\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\zeta }_{\nu }-{\eta }_{\nu }}{\epsilon ·{\left(\frac{1}{k}\right)}^{\frac{1}{{\mathtt{q}}^{-}}}}}\right)\le 1$, and we obtain

$$inf\left\{{\sigma }_{\nu };{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\zeta }_{\nu }-{\eta }_{\nu }}{\epsilon ·{\left({\sigma }_{\nu }\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\le {\displaystyle \frac{1}{\mathtt{k}}}$$

Therefore,

$$\begin{array}{cc}\hfill {\rho }_{\ell \mathtt{q}(·){log}^{\beta }{\mathtt{L}}^{\mathtt{p}(·)}}\left({\displaystyle \frac{\zeta -\eta }{\epsilon }}\right)& =\sum _{\nu =1}^{\infty }inf\left\{{\sigma }_{\nu }>0;{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\zeta }_{\nu }-{\eta }_{\nu }}{\epsilon {\sigma }_{\nu }^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\nu }-{\eta }_{\nu }|}{\epsilon {\sigma }_{\nu }^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\hfill \\ & =\sum _{\nu =1}^{\mathtt{k}}inf\left\{{\sigma }_{\nu }>0;{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\zeta }_{\nu }-{\eta }_{\nu }}{\epsilon {\sigma }_{\nu }^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\nu }-{\eta }_{\nu }|}{\epsilon {\sigma }_{\nu }^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\hfill \\ & \le 1.\hfill \end{array}$$

Now, we have show that

$$\overline{{\mathtt{p}}_{\mathtt{k}=1}^{\infty }}{{\mathcal{V}}_{\mathtt{k}}}^{{\parallel ·\parallel }_{\ell \mathtt{q}(·){log}^{\beta }{\mathtt{L}}^{\mathtt{p}(·)}}}={\ell }_{\mathtt{q}(·)}{log}^{\beta }{\mathtt{L}}^{\mathtt{p}(·)}.$$

To prove it, we let $\zeta \in {\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$, then

$$\sum _{\mathtt{i}=1}^{\infty }inf\left\{{\sigma }_{\mathtt{i}};{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{\zeta }{2\mathcal{M}{\sigma }_{\mathtt{i}}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|\zeta |}{2\mathcal{M}{\sigma }_{\mathtt{i}}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\le 1,$$

where $\mathcal{M}={\parallel \zeta \parallel }_{{\ell }_{\mathtt{q}(·)}{log}^{\beta }{\mathtt{L}}^{\mathtt{p}(·)}}$. Therefore, for every $0<\epsilon <1$, ∃$\mathtt{k}\in \mathtt{N}$ such that

$$\sum _{\mathtt{i}=\mathtt{k}+\mathtt{1}}^{\infty }inf\left\{{\sigma }_{\mathtt{i}};{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{\zeta }{2\mathcal{M}{\sigma }_{\mathtt{i}}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|\zeta |}{2\mathcal{M}{\sigma }_{\mathtt{i}}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\le \epsilon .$$

Thus, we obtain

$$\begin{array}{c}\hfill \sum _{\mathtt{i}=\mathtt{k}+\mathtt{1}}^{\infty }inf\left\{{\tau }_{\mathtt{i}};{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{\zeta }{2\mathcal{M}{\left(\epsilon {\tau }_{\mathtt{i}}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|\zeta |}{2\mathcal{M}{\left(\epsilon {\tau }_{\mathtt{i}}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\le 1.\end{array}$$

(21)

On the other hand, we have

$$\begin{array}{cc}\hfill & {\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{\zeta }{2\mathcal{M}{\epsilon }^{\frac{1}{{\mathtt{q}}^{+}}}{\tau }_{\mathtt{i}}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|\zeta |}{2\mathcal{M}{\epsilon }^{\frac{1}{{\mathtt{q}}^{+}}}{\tau }_{\mathtt{i}}^{\frac{1}{\mathtt{q}(·)}}}}\right)\hfill \\ & \le {\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{\zeta }{2\mathcal{M}{\left(\epsilon {\tau }_{\mathtt{i}}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|\zeta |}{2\mathcal{M}{\left(\epsilon {\tau }_{\mathtt{i}}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1.\hfill \end{array}$$

(22)

Thanks to (22), we have

$$\begin{array}{cc}\hfill & \left\{{\tau }_{\mathtt{i}};{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{\zeta }{2\mathcal{M}{\left(\epsilon {\tau }_{\mathtt{i}}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|\zeta |}{2\mathcal{M}{\left(\epsilon {\tau }_{\mathtt{i}}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\hfill \\ & \subset \left\{{\tau }_{\mathtt{i}};{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{\zeta }{2\mathcal{M}{\epsilon }^{\frac{1}{{\mathtt{q}}^{+}}}{\tau }_{\mathtt{i}}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|\zeta |}{2\mathcal{M}{\epsilon }^{\frac{1}{{\mathtt{q}}^{+}}}{\tau }_{\mathtt{i}}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}.\hfill \end{array}$$

(23)

Then, in view of (21), we have

$$\begin{array}{cc}\hfill \sum _{\mathtt{i}=\mathtt{k}+\mathtt{1}}^{\infty }inf& \left\{{\tau }_{\mathtt{i}};{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{\zeta }{2\mathcal{M}{\epsilon }^{\frac{1}{{\mathtt{q}}^{+}}}{\tau }_{\mathtt{i}}^{\left.\frac{1}{q·(}\right)}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|\zeta |}{2\mathcal{M}{\epsilon }^{\frac{1}{{\mathtt{q}}^{+}}}{\tau }_{\mathtt{i}}^{\left.\frac{1}{q·(}\right)}}}\right)\le 1\right\}\hfill \\ \hfill & \le \sum _{\mathtt{i}=\mathtt{k}+\mathtt{1}}^{\infty }inf\left\{{\tau }_{\mathtt{i}};{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{\zeta }{2\mathcal{M}{\left(\epsilon {\tau }_{\mathtt{i}}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|\zeta |}{2\mathcal{M}{\left(\epsilon {\tau }_{\mathtt{i}}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\le 1\hfill \end{array}$$

(24)

Finally, let us define

$$\eta =\left\{\begin{array}{cc}{\zeta }_{\mathtt{i}}\hfill & \mathrm{if}i\le \mathtt{k}\hfill \\ 0\hfill & \mathrm{if}\mathtt{i}>\mathtt{k}\hfill \end{array}.\right.$$

It is clear that $\zeta \in {\mathcal{V}}_{\mathtt{k}}$, and by (24), we obtain ${\parallel \zeta -\eta \parallel }_{{\ell }_{\mathtt{q}(·)}{log}^{\beta }{\mathtt{L}}^{\mathtt{p}(·)}}\le 2\mathcal{M}{\epsilon }^{{\displaystyle \frac{1}{{\mathtt{q}}^{+}}}}$, which is the desired result. □

Proposition 3. 

Let $\mathtt{p},\mathtt{q}\in {\mathcal{P}}_0\left(\Omega \right)$ and let $\left({\zeta }_{\mathtt{j}}\right),\left({\eta }_{\mathtt{j}}\right)\subset {\mathtt{L}}^{\mathtt{p}(·)}\left({\mathtt{R}}^n\right)$. If $0\le {\zeta }_{\mathtt{j}}\le {\eta }_{\mathtt{j}}$ for each $\mathtt{j}\in \mathtt{N}$, then

$${∥\left({\zeta }_{\mathtt{j}}\right)∥}_{{\ell }_{\mathtt{q}(·)}{log}^{\beta }{\mathtt{L}}^{\mathtt{p}(·)}}\le {∥\left({\eta }_{\mathtt{j}}\right)∥}_{{\ell }_{\mathtt{q}(·)}{log}^{\beta }{\mathtt{L}}^{\mathtt{p}(·)}}.$$

It suffices to show that

$$\begin{array}{cc}\hfill & \left\{\sigma >0:{\rho }_{\ell \mathtt{q}(·)\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\left({\displaystyle \frac{{\eta }_{\mathtt{j}}}{\sigma }}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\eta }_{\mathtt{j}}|}{\sigma }}\right)\le 1\right\}\hfill \\ & \subset \left\{\sigma >0:{\rho }_{\ell \mathtt{q}(·)\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\left({\displaystyle \frac{{\zeta }_{\mathtt{j}}}{\sigma }}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mathtt{j}}|}{\sigma }}\right)\le 1\right\}.\hfill \end{array}$$

(25)

Suppose that $\sigma >0$ is such that

$${\rho }_{{\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\left({\displaystyle \frac{{\eta }_{\mathtt{j}}}{\sigma }}\right)=\sum _{\mathtt{j}=\mathtt{1}}^{\infty }inf\left\{{\tau }_{\mathtt{j}}>0:{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\eta }_{\mathtt{j}}}{\sigma {\tau }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\eta }_{\mathtt{j}}|}{\sigma {\tau }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\le 1.$$

Let us fix $\mathtt{j}\in \mathtt{N}$ and let ${\tau }_{\mathtt{j}}>0$ be such that

$${\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\eta }_{\mathtt{j}}}{\sigma {\tau }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\eta }_{\mathtt{j}}|}{\sigma {\tau }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1.$$

Due to assumption that $0\le {\zeta }_{\mathtt{j}}\le {\eta }_{\mathtt{j}}$, we obtain

$${\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\zeta }_{\mathtt{j}}}{\sigma {\tau }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mathtt{j}}|}{\sigma {\tau }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le {\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\eta }_{\mathtt{j}}}{\sigma {\tau }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\eta }_{\mathtt{j}}|}{\sigma {\tau }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1.$$

Therefore,

$$\begin{array}{cc}\hfill & \left\{{\tau }_{\mathtt{j}}>0:{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\eta }_{\mathtt{j}}}{\sigma {\tau }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\eta }_{\mathtt{j}}|}{\sigma {\tau }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\hfill \\ & \subset \left\{{\tau }_{\mathtt{j}}>0:{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\eta }_{\mathtt{j}}}{\sigma {\tau }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\eta }_{\mathtt{j}}|}{\sigma {\tau }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}.\hfill \end{array}$$

(26)

Finally, it easily follows that

$$\begin{array}{cc}\hfill {\rho }_{{\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\left({\displaystyle \frac{{\zeta }_{\mathtt{j}}}{\sigma }}\right)& =\sum _{\mathtt{j}=\mathtt{1}}^{\infty }inf\left\{{\tau }_{\mathtt{j}}>0:{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\zeta }_{\mathtt{j}}}{\sigma {\tau }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mathtt{j}}|}{\sigma {\tau }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\hfill \\ & \le \sum _{\mathtt{j}=\mathtt{1}}^{\infty }inf\left\{{\tau }_{\mathtt{j}}>0:{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\eta }_{\mathtt{j}}}{\sigma {\tau }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\eta }_{\mathtt{j}}|}{\sigma {\tau }_{\mathtt{j}}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\hfill \\ & ={\rho }_{{\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\left({\displaystyle \frac{{\eta }_{\mathtt{j}}}{\sigma }}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\eta }_{\mathtt{j}}|}{\sigma }}\right)\le 1,\hfill \end{array}$$

and this concludes the proof.

Lemma 5. 

Assume that $\mathtt{p},\mathtt{q}\in {\mathcal{P}}_0\left(\Omega \right)$ with ${\left({\zeta }_{\mathtt{i}}\right)}_{\mathtt{i}=1}^{\infty }\in {\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$. Then, for every $\mathtt{j}\in \mathtt{N}$, we have

$${∥{\zeta }_{\mathtt{j}}∥}_{{\mathtt{L}}^{\mathtt{p}(·)}\left(\Omega \right)}\le {∥\left({\zeta }_{\mathtt{j}}\right)∥}_{{\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}.$$

Proof. 

Suppose $\mathtt{j}\in \mathtt{N}$, and we define the sequence ${\left({\eta }_{\mathtt{i}}\left(\mathtt{j}\right)\right)}_{\mathtt{i}=\mathtt{1}}^{\infty }$ in such a way that ${\eta }_{\mathtt{i}}\left(\mathtt{j}\right):={\zeta }_{\mathtt{j}}{\delta }_{\mathtt{i}\mathtt{j}}$. Then, for each $\mathtt{i}\in \mathtt{N},$ we have $\left|{\eta }_{\mathtt{i}}\left(\mathtt{j}\right)\right|\le \left|{\zeta }_{\mathtt{j}}\right|$, as sequences preserve their order, it follows as

$${∥\left({\eta }_{\mathtt{j}}\left(\mathtt{j}\right)\right)∥}_{{\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\le {∥\left({\zeta }_{\mathtt{j}}\right)∥}_{{\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}.$$

Finally, by Proposition 3, we obtain

$${∥\left({\eta }_{\mathtt{j}}\left(\mathtt{j}\right)\right)∥}_{{\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}={∥{\zeta }_{\mathtt{j}}∥}_{{\mathtt{L}}^{\mathtt{p}(·)}},$$

and the proof follows. □

5.3. Precompactness

Theorem 13. 

Let $\mathtt{p}\in {\mathcal{P}}_0^{log}\left(\Omega \right)$, $\mathtt{q}\in {\mathcal{P}}_0\left(\Omega \right)$ and assume that ${\mathtt{p}}_{+},{\mathtt{q}}_{+}<\infty$. Then, the class $\mathcal{Q}\subset {\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$ is precompact in ${\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$ if the following assumptions are met:

• (a) $\mathcal{Q}$ is precompact in ${\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$;
• (b) For all $\epsilon >0$ there exists $\mathcal{K}\in \mathtt{N}$ such that ${\forall }_{\zeta \in \mathcal{Q}}{\sum }_{\mathtt{r}=\mathcal{K}+\mathtt{1}}^{\infty }{∥{\left|{\zeta }_{\mathtt{r}}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}\left(\Omega \right)}\le \epsilon ;$
• (c) For all $\mathtt{r}\in \mathtt{N}$ and $0<\mathtt{u}<{\mathtt{p}}_{-}$, then

  $$\underset{\sigma \to 0^{+}}{lim}\underset{\zeta \in \mathcal{Q}}{sup}{\int }_{\Omega }{\left({\displaystyle \frac{1}{\left|\mathcal{B}(\mu ,\sigma )\right|}}{\int }_{\mathtt{B}(\mu ,\sigma )}{\left|{\zeta }_{\mathtt{r}}(\mu )-{\zeta }_{\mathtt{r}}(\nu )\right|}^{\mathtt{u}}\mathtt{d}\nu \right)}^{{\displaystyle \frac{\mathtt{p}(\mu )}{\mathtt{u}}}}\mathtt{d}\mu =0;$$
• (d) For all $\mathtt{r}\in \mathtt{N},$${lim}_{\mathtt{R}\to \infty }{sup}_{\zeta \in \mathcal{Q}}{\int }_{\Omega \setminus \mathcal{B}(0,\mathtt{R})}{\left|{\zeta }_{\mathtt{r}}(\mu )\right|}^{\mathtt{p}(\mu )}\mathtt{d}\mu =0$.

Proof. 

Initially, we will demonstrate that the conditions $\left(a\right)-\left(d\right)$ are satisfied. The space ${\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$ is a quasi-Banach space; it is sufficient to establish that $\left({\zeta }^{\mathtt{k}}\right)$ has a convergent subsequence.

For every $\mathtt{r}\in \mathtt{N}$ sequence, ${\left({\zeta }_{\mathtt{r}}^{\mathtt{k}}\right)}_{\mathtt{k}=1}^{\infty }$ is bounded in ${\mathtt{L}}^{\mathtt{p}(·)}(\Omega )$. Given $\left(a\right)$, there exists $\alpha \in \mathtt{R}$ such that

$$\underset{\mathtt{k}\in \mathtt{N}}{sup}\left\{\mathtt{k}:{\rho }_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\left({\zeta }^{\mathtt{k}}\right){log}^{\beta }\left(e+|{\zeta }^{\mathtt{k}}|\right)\le \alpha \right\},$$

that is

$$\underset{\mathtt{k}\in \mathtt{N}}{sup}{∥{\zeta }^{\mathtt{k}}∥}_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\le \alpha .$$

Let us fix $\mathtt{r}\in \mathtt{N}$. Using Lemma 5, we may conclude that

$$\underset{\mathtt{k}\in \mathtt{N}}{sup}\left\{\mathtt{k}:{\rho }_{{\mathtt{L}}^{\mathtt{p}(·)}}\left({{\zeta }_{\mathtt{r}}}^{\mathtt{k}}\right){log}^{\beta }\left(e+|{{\zeta }_{\mathtt{r}}}^{\mathtt{k}}|\right)\le \alpha \right\}\le \underset{\mathtt{k}\in \mathtt{N}}{sup}\left\{\mathtt{k}:{\rho }_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\left({\zeta }^{\mathtt{k}}\right){log}^{\beta }\left(e+|{\zeta }^{\mathtt{k}}|\right)\le \alpha \right\},$$

and this implies that

$$\underset{\mathtt{k}\in \mathtt{N}}{sup}{∥{\zeta }_{\mathtt{r}}^{\mathtt{k}}∥}_{{\mathtt{L}}^{\mathtt{p}(·)}(\Omega )}\le \underset{\mathtt{k}\in \mathtt{N}}{sup}{∥{\zeta }^{\mathtt{k}}∥}_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\le \alpha ,$$

which proves that ${\left({\zeta }_{\mathtt{r}}^{\mathtt{k}}\right)}_{\mathtt{k}=\mathtt{1}}^{\infty }$ is bounded in ${\mathtt{L}}^{\mathtt{p}(·)}(\Omega ).$ Finally, for a constant $\mathtt{r}\in \mathtt{N}$, we define

$${\mathcal{K}}_{\mathtt{r}}:=\overline{\left\{{\zeta }_{\mathtt{r}}^{\mathtt{k}}\in {\mathtt{L}}^{\mathtt{p}(·)}(\Omega ):\mathtt{k}\in \mathtt{N}\right\}}.$$

Since sets ${\left({\mathcal{K}}_{\mathtt{r}}\right)}_{\mathtt{j}=\mathtt{1}}^{\infty }$ are closed and bounded in ${\mathtt{L}}^{\mathtt{p}(·)}(∆)$, thus, a strictly increasing sequence $\left({\mathtt{k}}_{\mathtt{l}}\right)\subset \mathtt{N}$ and a sequence $\left({\zeta }_{\mathtt{r}}\right)\subset {\mathtt{L}}^{\mathtt{p}(·)}(\Omega )$ exist by using the Cantor diagonal argument technique for every $\mathtt{r}\in \mathtt{N}$, that is

$${\zeta }_{\mathtt{r}}^{{\mathtt{k}}_{\mathtt{l}}}\stackrel{l\to \infty }{\to }{\zeta }_{\mathtt{r}}\mathrm{in}{\mathtt{L}}^{\mathtt{p}(·)}(\Omega ).$$

Let $\zeta =\left({\zeta }_{\mathtt{r}}\right)$, and we prove

$${\zeta }^{{\mathtt{k}}_{\mathtt{l}}}\stackrel{l\to \infty }{\to }\zeta in{\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right).$$

We denote $\left({\zeta }^{{\mathtt{k}}_{\mathtt{l}}}\right)$ by $\left({\zeta }^{\mathtt{k}}\right)$. Fixing $\epsilon \in (0,1)$, we demonstrate that

$${\rho }_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\left({\displaystyle \frac{{\zeta }^{\mathtt{k}}-\zeta }{\epsilon }}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }^{\mathtt{k}}-\zeta |}{\epsilon }}\right)\le 1,$$

for sufficiently large $\mathtt{k}\in \mathtt{N}$. There exists $\mathcal{K}\in \mathtt{N}$ from $\left(b\right)$ such that

$$\begin{array}{c}\hfill \sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }{∥{\left|{\zeta }_{\mathtt{r}}^{\mathtt{k}}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}\le {\displaystyle \frac{{\epsilon }^{{\mathtt{q}}_{+}}}{2^{{\mathtt{q}}_{+}}4\mathcal{L}{\left({log}^{\beta }\right)}^{{\mathtt{q}}_{+}}}}\le \epsilon .\end{array}$$

(27)

Subsequently, we can split the modular as follows:

$$\begin{array}{cc}\hfill & {\rho }_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\left({\displaystyle \frac{{\zeta }^{\mathtt{k}}-\zeta }{\epsilon }}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }^{\mathtt{k}}-\zeta |}{\epsilon }}\right)\hfill \\ & =\sum _{\mathtt{j}=\mathtt{1}}^{\mathcal{K}}{∥{\left|{\displaystyle \frac{{\zeta }_{\mathtt{r}}^{\mathtt{k}}-{\zeta }_{\mathtt{r}}}{\epsilon }}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}+\sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }{∥{\left|{\displaystyle \frac{{\zeta }_{\mathtt{r}}^{\mathtt{k}}-{\zeta }_{\mathtt{r}}}{\epsilon }}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}\hfill \end{array}$$

(28)

Next, by the triangle inequality for ${\parallel ·\parallel }_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}$, Lemma 5, the Fatou lemma for the counting measure, and Equation (27), we have

$$\begin{array}{cc}\hfill & \sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }{∥{\left|{\zeta }_{\mathtt{r}}^{\mathtt{k}}-{\zeta }_{\mathtt{r}}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}\le \sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }{∥2^{{\mathtt{q}}_{+}}\left({\left|{\displaystyle \frac{{\zeta }_{\mathtt{r}}^{\mathtt{k}}}{\epsilon }}\right|}^{\mathtt{q}(·)}+{\left|{\displaystyle \frac{{\zeta }_{\mathtt{r}}}{\epsilon }}\right|}^{\mathtt{q}(·)}\right)∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}(\Omega )}}\hfill \\ \hfill & \le 2^{{\mathtt{q}}_{+}}\mathcal{L}\left(\sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }{∥{\left|{\displaystyle \frac{{\zeta }_{\mathtt{r}}^{\mathtt{k}}}{\epsilon }}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}(\Omega )}}+\sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }{∥{\left|{\displaystyle \frac{{\zeta }_{\mathtt{r}}}{\epsilon }}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}\right)\hfill \\ \hfill & \le {\displaystyle \frac{2^{{\mathtt{q}}_{+}}\mathcal{L}}{{\epsilon }^{{\mathtt{q}}_{+}}}}\left(\sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }{∥{\left|{\zeta }_{\mathtt{r}}^{\mathtt{k}}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}+\sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }{∥{\left|{\zeta }_{\mathtt{r}}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}\right)\hfill \\ & ={\displaystyle \frac{2^{{\mathtt{q}}_{+}}\mathcal{L}}{{\epsilon }^{{\mathtt{q}}_{+}}}}\left(\sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }{∥{\left|{\zeta }_{\mathtt{r}}^{\mathtt{k}}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}+\sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }inf\left\{{\tau }_{\mathtt{r}}>0:{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\zeta }_{\mathtt{r}}}{{\tau }_{\mathtt{r}}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mathtt{r}}|}{{\tau }_{\mathtt{r}}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\right)\hfill \\ & \le {\displaystyle \frac{2^{{\mathtt{q}}_{+}}4\mathcal{L}{\left({log}^{\beta }\right)}^{{\mathtt{q}}_{+}}}{{\epsilon }^{{\mathtt{q}}_{+}}}}\left(\sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }{∥{\left|{\zeta }_{\mathtt{r}}^{\mathtt{k}}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}+\sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }\underset{\mathtt{k}\to \infty }{lim\; inf}inf\left\{{\tau }_{\mathtt{r}}\left(\mathtt{k}\right)>0:{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\zeta }_{\mathtt{r}}^{\mathtt{k}}}{{\tau }_{\mathtt{r}}{\left(\mathtt{k}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mathtt{r}}^{\mathtt{k}}|}{{\tau }_{\mathtt{r}}{\left(\mathtt{k}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\right)\hfill \\ & \le {\displaystyle \frac{2^{{\mathtt{q}}_{+}}4\mathcal{L}{\left({log}^{\beta }\right)}^{{\mathtt{q}}_{+}}}{{\epsilon }^{{\mathtt{q}}_{+}}}}\left(\sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }{∥{\left|{\zeta }_{\mathtt{r}}^{\mathtt{k}}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}+\underset{\mathtt{k}\to \infty }{lim\; inf}\sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }inf\left\{{\tau }_{\mathtt{r}}\left(k\right)>0:{\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\zeta }_{\mathtt{r}}^{\mathtt{k}}}{{\tau }_{\mathtt{r}}{\left(\mathtt{k}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mathtt{r}}^{\mathtt{k}}|}{{\tau }_{\mathtt{r}}{\left(\mathtt{k}\right)}^{\frac{1}{\mathtt{q}(·)}}}}\right)\le 1\right\}\right)\hfill \\ & ={\displaystyle \frac{2^{{\mathtt{q}}_{+}}4\mathcal{L}{\left({log}^{\beta }\right)}^{{\mathtt{q}}_{+}}}{{\epsilon }^{{\mathtt{q}}_{+}}}}\left(\sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }{∥{\left|{\zeta }_{\mathtt{r}}^{\mathtt{k}}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}+\underset{\mathtt{k}\to \infty }{lim\; inf}\sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }{∥{\left|{\zeta }_{\mathtt{r}}^{\mathtt{k}}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}\right)\hfill \\ & \le {\displaystyle \frac{2^{{\mathtt{q}}_{+}}4\mathcal{L}{\left({log}^{\beta }\right)}^{{\mathtt{q}}_{+}}}{{\epsilon }^{{\mathtt{q}}_{+}}}}·{\displaystyle \frac{{\epsilon }^{{\mathtt{q}}_{+}}}{2^{{\mathtt{q}}_{+}+2}4\mathcal{L}{\left({log}^{\beta }\right)}^{{\mathtt{q}}_{+}}}}={\displaystyle \frac{1}{4}}\le \epsilon .\hfill \end{array}$$

Therefore, for each $\mathtt{k}\in \mathtt{N}$, we have

$$\begin{array}{c}\hfill {\rho }_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\left({\displaystyle \frac{{\zeta }^{\mathtt{k}}-\zeta }{\epsilon }}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }^{\mathtt{k}}-\zeta |}{\epsilon }}\right)\le \sum _{\mathtt{j}=\mathtt{1}}^{\mathcal{K}}{∥{\left|{\displaystyle \frac{{\zeta }_{\mathtt{r}}^{\mathtt{k}}-{\zeta }_{\mathtt{r}}}{\epsilon }}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}+{\displaystyle \frac{1}{4}}.\end{array}$$

(29)

Additionally, we assert that for any $\mathtt{r}\in \{1,\dots ,\mathcal{K}\}$, we also have

$${∥{\left|{\zeta }_{\mathtt{r}}^{\mathtt{k}}-{\zeta }_{\mathtt{r}}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}\stackrel{\mathtt{k}\to \infty }{⟶}0.$$

And for some specific $\mathtt{r}\in \{1,\dots ,\mathcal{K}\}$, and $\nu \in (0,1)$, we arrive at

$$\begin{array}{cc}\hfill & {\rho }_{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}\left({\left|{\displaystyle \frac{{\zeta }_{\mathtt{r}}^{\mathtt{k}}-{\zeta }_{\mathtt{r}}}{\epsilon }}\right|}^{\mathtt{q}(·)}·{\displaystyle \frac{1}{\nu }}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mathtt{r}}^{\mathtt{k}}-{\zeta }_{\mathtt{r}}|}{\epsilon \nu }}\right)\hfill \\ & \le {\int }_{\Omega }{\left|{\displaystyle \frac{{\zeta }_{\mathtt{r}}^{\mathtt{k}}\left(\mathtt{x}\right)-{\zeta }_{\mathtt{r}}\left(\mathtt{x}\right)}{\epsilon {\nu }^{\frac{1}{{\mathtt{q}}_{-}}}}}\right|}^{\mathtt{p}\left(\mathtt{x}\right)}{log}^{\beta }\left(e+\left|{\displaystyle \frac{{\zeta }_{\mathtt{r}}^{\mathtt{k}}\left(\mathtt{x}\right)-{\zeta }_{\mathtt{r}}\left(\mathtt{x}\right)}{\epsilon {\nu }^{\frac{1}{{\mathtt{q}}_{-}}}}}\right|\right)\mathrm{dx}\hfill \\ & ={\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\zeta }_{\mathtt{r}}^{\mathtt{k}}-{\zeta }_{\mathtt{r}}}{\epsilon {\nu }^{\frac{1}{{\mathtt{q}}_{-}}}}}\right){log}^{\beta }\left(e+\left|{\displaystyle \frac{{\zeta }_{\mathtt{r}}^{\mathtt{k}}\left(\mathtt{x}\right)-{\zeta }_{\mathtt{r}}\left(\mathtt{x}\right)}{\epsilon {\nu }^{\frac{1}{{\mathtt{q}}_{-}}}}}\right|\right).\hfill \end{array}$$

(30)

Since ${\zeta }_{\mathtt{r}}^{\mathtt{k}}\stackrel{\mathtt{k}\to \infty }{\to }{\zeta }_{\mathtt{r}}$ in ${\mathtt{L}}^{\mathtt{p}(·)}(\Omega )$, for sufficiently big $\mathtt{k}\in \mathtt{N}$, we have

$$\begin{array}{cc}\hfill & {\rho }_{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}\left({\left|{\displaystyle \frac{{\zeta }_{\mathtt{r}}^{\mathtt{k}}-{\zeta }_{\mathtt{r}}}{\epsilon }}\right|}^{\mathtt{q}(·)}·{\displaystyle \frac{1}{\nu }}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }_{\mathtt{r}}^{\mathtt{k}}-{\zeta }_{\mathtt{r}}|}{\epsilon \nu }}\right)\hfill \\ & \le {\rho }_{\mathtt{p}(·)}\left({\displaystyle \frac{{\zeta }_{\mathtt{r}}^{\mathtt{k}}-{\zeta }_{\mathtt{r}}}{\epsilon {\nu }^{\frac{1}{{\mathtt{q}}_{-}}}}}\right){log}^{\beta }\left(e+\left|{\displaystyle \frac{{\zeta }_{\mathtt{r}}^{\mathtt{k}}\left(\mathtt{x}\right)-{\zeta }_{\mathtt{r}}\left(\mathtt{x}\right)}{\epsilon {\nu }^{\frac{1}{{\mathtt{q}}_{-}}}}}\right|\right)\le 1,\hfill \end{array}$$

(31)

which proves that

$${∥{\left|{\displaystyle \frac{{\zeta }_{\mathtt{r}}^{\mathtt{k}}-{\zeta }_{\mathtt{r}}}{\epsilon }}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}\stackrel{\mathtt{k}\to \infty }{\to }0.$$

According to Equation (29), for sufficiently large $\mathtt{k}\in \mathtt{N}$, we have

$${\rho }_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\left({\displaystyle \frac{{\zeta }^{\mathtt{k}}-\zeta }{\epsilon }}\right){log}^{\beta }\left(e+{\displaystyle \frac{|{\zeta }^{\mathtt{k}}-\zeta |}{\epsilon }}\right)\le 1,$$

this indicates that ${\zeta }^{\mathtt{k}}\stackrel{\mathtt{k}\to \infty }{\to }\zeta$ in $\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$ and it proves that the class $\mathcal{Q}$ of measurable functions is bounded. Then, we prove that conditions $\left(a\right)-\left(d\right)$ are necessary, assuming $\mathcal{Q}$ is totally bounded. Since $\left(a\right)$ is evident, we shift our focus to $\left(b\right)$. Let us set $\epsilon \in (0,1)$ and define

$$\kappa :={\displaystyle \frac{\epsilon }{2^{{\mathtt{q}}_{+}+1}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }}},$$

where the constant $\mathcal{L}$ is once more used for the triangle inequality ${∥{\left|{\zeta }_{\mathtt{r}}^{\mathtt{k}}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}$. According to our assumptions, there exists ${\zeta }^1,\dots ,{\zeta }^{\mathtt{k}}\in \mathcal{Q}$ such that

$$\mathcal{Q}\subset {\mathtt{p}}_{\mathtt{i}=\mathtt{1}}^{\mathtt{k}}\mathcal{B}\left({\zeta }^{\mathtt{i}},{\left({\displaystyle \frac{\epsilon }{2^{{\mathtt{q}}_{+}+1}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }}}\right)}^{{\displaystyle \frac{1}{{\mathtt{q}}_{-}}}}\right),$$

where $\mathcal{B}\left({\zeta }^{\mathtt{i}},{\left({\displaystyle \frac{\epsilon }{2^{{\mathtt{q}}_{+}+1}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }}}\right)}^{{\displaystyle \frac{1}{{\mathtt{q}}_{-}}}}\right)$ are an open balls in $\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$.

Since ${\zeta }_i\in \ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$, ∃$\mathcal{K}\in \mathtt{N}$ such that

$$\sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }{∥{\left|{\zeta }_{\mathtt{r}}^{\mathtt{i}}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}<{\displaystyle \frac{\epsilon }{2^{{\mathtt{q}}_{+}+1}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }}},$$

for $\mathtt{i}=1,\dots ,\mathtt{k}$. Let $\zeta \in \mathcal{Q}$, then ∃$\mathtt{i}\in \{1,\dots ,\mathtt{k}\}$, such that

$${∥\zeta -{\zeta }^i∥}_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}<{\left({\displaystyle \frac{\epsilon }{2^{{\mathtt{q}}_{+}+1}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }}}\right)}^{{\displaystyle \frac{1}{{\mathtt{q}}_{-}}}}.$$

If we consider the unit ball property, then it follows that

$${\rho }_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\left({\displaystyle \frac{\zeta -{\zeta }^i}{{\left(\frac{\epsilon }{2^{{\mathtt{q}}_{+}+1}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }}\right)}^{\frac{1}{{\mathtt{q}}_{-}}}}}\right){log}^{\beta }\left(e+\left|{\displaystyle \frac{\zeta -{\zeta }^i}{{\left(\frac{\epsilon }{2^{{\mathtt{q}}_{+}+1}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }}\right)}^{\frac{1}{{\mathtt{q}}_{-}}}}}\right|\right)\le 1.$$

Finally, the triangle inequality yields

$$\begin{array}{cc}\hfill & \sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }{∥{\left|{\zeta }_{\mathtt{r}}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}\hfill \\ \hfill & \le 2^{{\mathtt{q}}_{+}}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }\left(\sum _{\mathtt{j}=1}^{\infty }{∥{\left|{\zeta }_{\mathtt{r}}-{\zeta }_{\mathtt{r}}^{\mathtt{i}}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}+\sum _{\mathtt{r}=\mathcal{K}+1}^{\infty }{∥{\left|{\zeta }_{\mathtt{r}}^{\mathtt{i}}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}\right)\hfill \\ \hfill & \le 2^{{\mathtt{q}}_{+}}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }\left({\displaystyle \frac{\epsilon }{2^{{\mathtt{q}}_{+}+1}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }}}\sum _{\mathtt{r}=1}^{\infty }{\displaystyle \frac{2^{{\mathtt{q}}_{+}+1}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }}{\epsilon }}{∥{\left|{\zeta }_{\mathtt{r}}-{\zeta }_{\mathtt{r}}^i\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}+{\displaystyle \frac{\epsilon }{2^{{\mathtt{q}}_{+}+1}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }}}\right)\hfill \\ \hfill & \le 2^{{\mathtt{q}}_{+}}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }\left({\displaystyle \frac{\epsilon }{2^{{\mathtt{q}}_{+}+1}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }}}\sum _{\mathtt{r}=1}^{\infty }{∥{\left|{\displaystyle \frac{{\zeta }_{\mathtt{r}}-{\zeta }_{\mathtt{r}}^i}{{\kappa }^{\frac{1}{{\mathtt{q}}_{-}}}}}\right|}^{\mathtt{q}(·)}∥}_{{\mathtt{L}}^{{\displaystyle \frac{\mathtt{p}(·)}{\mathtt{q}(·)}}}(\Omega )}+{\displaystyle \frac{\epsilon }{2^{{\mathtt{q}}_{+}+1}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }}}{\left({log}^{\beta }\right)}^{\epsilon }\right)\hfill \\ \hfill & =2^{{\mathtt{q}}_{+}}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }\left({\displaystyle \frac{\epsilon }{2^{{\mathtt{q}}_{+}+1}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }}}{\rho }_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)}\left({\displaystyle \frac{\zeta -{\zeta }^i}{{\left(\frac{\epsilon }{2^{{\mathtt{q}}_{+}+1}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }}\right)}^{\frac{1}{{\mathtt{q}}_{-}}}}}\right)\right.\hfill \\ \hfill & \times {log}^{\beta }\left(e+\left|{\displaystyle \frac{\zeta -{\zeta }^i}{{\left(\frac{\epsilon }{2^{{\mathtt{q}}_{+}+1}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }}\right)}^{\frac{1}{{\mathtt{q}}_{-}}}}}\right|\right)+{\displaystyle \frac{\epsilon }{2^{{\mathtt{q}}_{+}+1}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }}}{\left({log}^{\beta }\right)}^{\epsilon })\hfill \\ \hfill & \le 2^{{\mathtt{q}}_{+}+1}\mathcal{L}{\left({log}^{\beta }\right)}^{\epsilon }\kappa =\epsilon .\hfill \end{array}$$

which establishes $\left(b\right)$.

To show that conditions $\left(c\right)$ and $\left(d\right)$ are valid, let us consider $\mathtt{r}\in \mathtt{N}$ and analyze Lemma 5, then one has

$${∥{\zeta }_{\mathtt{r}}∥}_{{\mathtt{L}}^{\mathtt{p}(·)}(\Omega )}\le {\parallel \zeta \parallel }_{\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)},$$

valid for each $\zeta \in \ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$. As a result, the projective map

$${\pi }_{\mathtt{r}}:\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)\to {\mathtt{L}}^{\mathtt{p}(·)}(\Omega ),$$

is defined as follows:

$${\pi }_{\mathtt{r}}(\zeta ):={\zeta }_{\mathtt{r}}\mathrm{for}\zeta \in \ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right).$$

As $\ell {\mathtt{q}}^{(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$ is a quasi-Banach space, it is a closed and bounded subset of $\mathcal{Q}$, which is equivalent to its precompactness. Following that, the continuous image of the projective map is defined as

$${\pi }_{\mathtt{r}}\left(\mathcal{Q}\right)=\left\{{\zeta }_{\mathtt{r}}:\zeta \in \mathcal{Q}\right\},$$

which is precompact in ${\mathtt{L}}^{\mathtt{p}(·)}(\Omega )$, and the proof is completed.

The results of Theorem 13 show that if $\mathtt{p}\in {\mathcal{P}}_0(\Omega )$ instead of $\mathtt{p}\in {\mathcal{P}}_0^{log}(\Omega )$, assumptions $\left(a\right)-\left(d\right)$ imply the closedness and boundedness. As a result, the theorem holds true. □

6. Conclusions and Future Remarks

The structure of function spaces and the behavior of operators are clarified by the equalities and inequalities of the Hilbert space, which are essential for functional analysis. The paper contributes in two important ways. The first is the employment of continuous self-adjoint operators on Hilbert spaces, which are essential when working with complex functions on higher dimensions, to extend equality from the classical sense to the tensorial sense using the tensor approximation. Next, the authors introduce a new type of function space, namely mixed-Orlicz–Zygmund space, which combines the Orlicz–Zygmund space of integrability with the sequence space. To prove its validity, we also examine several of its characteristics, including the completeness, compactness, and separability. We believe that this generalized space plays a very essential role in the progress of function spaces and their linked problems, such as the bounds of operators, the regularity of partial differential equations, embedding, etc.

This study advances mathematical inequality theory by investigating inequalities supporting tensor Hilbert spaces and mixed norm function spaces, which is a rare topic in the literature in terms of produced results. Following these results, we advise readers to attempt to develop the Maclaurin inequality on coordinates using tensor operations and also investigate various other properties of our newly defined mixed spaces, such as checking the boundedness of operators, analysis of differential equations, extending over a complex domain, etc.