Exploration of Hermite–Hadamard-Type Integral Inequalities for Twice Differentiable h-Convex Functions

Abstract

The significance of fractional calculus cannot be underestimated, as it plays a crucial role in the theory of inequalities. In this paper, we study a new class of mean-type inequalities by incorporating Riemann-type fractional integrals. By doing so, we discover a novel set of such inequalities and analyze them using different mathematical identities. This particular class of inequalities is introduced by employing a generalized convexity concept. To validate our work, we create visual graphs and a table of values using specific functions to represent the inequalities. This approach allows us to demonstrate the validity of our findings and further solidify our conclusions. Moreover, we find that some previously published results emerge as special consequences of our main findings. This research serves as a catalyst for future investigations, encouraging researchers to explore more comprehensive outcomes by using generalized fractional operators and expanding the concept of convexity.

1. Introduction and Preliminaries

Fractional calculus deals with arbitrary order integrals and derivatives that are employed in several applications. In recent years, the area related to fractional differential and integral equations has received much attention from numerous mathematicians and specialists [1]. The derivatives of fractional order represent physical models of multiple phenomena in many fields, including engineering [2,3], mathematical physics [4], fractional calculus [5] and bio-engineering [6]. The idea of convexity has been modernized, extended and expanded in several ways [7,8]. Convexity has a significant impact on our daily lives because of its numerous applications in commerce [9], industry [10], medicine [11] and the arts [12]. Geometrically convex functions in n-dimensions and s-dimensions have been derived. Some classes of convex functions, such as geometrically convex and s-convex, were investigated by Yang and Hudzik et al. in [13,14], respectively. Functional analysis, optimization and control theory all depend on convexity in significant ways [15]. Due to the activities of its definition, convexity has a strong character in the area of inequalities. The idea of convexity has played a significant and astonishing role in integral inequalities that are equally important for both fields of pure and applied mathematics [16,17]. It is obvious that without inequalities, mathematical techniques are meaningless. The mathematician presented extensions, refinements and modifications of classical inequalities such as Hermite–Hadamard inequalities [18,19], Ostrowski inequalities [20], Olsen inequalities [21], Gagliardo–Nirenberg inequalities [22] and Hardy-type inequalities [23].

The classical Hermite–Hadamard inequality was first introduced in 1893 [24]. Peter Korus [25,26] successfully updated a class of Hermite–Hadamard inequalities by incorporating the class of generalized convex derivatives. Some other authors studied the Fejer–Hadamard inequalities for convex functions in [27,28,29,30,31,32]. This inequality basically provides the bounds of the average value of a convex function. Moreover, it provides error boundaries of specific means relations and numerous numerical quadrature rules of integration such as rectangular, trapezoidal and Simpson [33,34]. For further studies, we refer the reader to articles [35,36] and book [37]. In the proposed work, we first establish some identities involving fractional integrals of the Riemann-type. Such identities will then be used to investigate Hermite–Hadamard-type integral inequalities for twice differentiable h-convex functions. The Hölders inequality will be utilized to create this class. The gamma function is defined in [38] by the following.

Definition 1.

The Euler gamma function defined for $Re\left(y\right)>0$, as

$$\begin{array}{c}\hfill \Gamma \left(y\right)={\int }_0^{\infty }{\xi }^{y-1}{e}^{-\xi }d\xi .\end{array}$$

In [39], Mubeen et al. defined the following definition of the k-gamma function.

Definition 2.

The k-gamma functions defined for $Re\left(y\right)>0$, as

$$\begin{array}{c}\hfill {\Gamma }_k\left(y\right)=\underset{m\to \infty }{lim}\frac{m!k^m{\left(mk\right)}^{\frac{y}{k}-1}}{{\left(y\right)}_{m,k}},\end{array}$$

where ${\left(y\right)}_{m,k}$ is the Pochhammer k-symbols for $k>0$ and factorial function. The integral form is given by

$$\begin{array}{c}\hfill {\Gamma }_k\left(y\right)={\int }_0^{\infty }{\xi }^{y-1}{exp}^{(-\frac{{\xi }^k}{k})}d\xi .\end{array}$$

Clearly, $\Gamma \left(y\right)={lim}_{k\to 1}{\Gamma }_k\left(y\right)$, and the relation between classical gamma and k-gamma is ${\Gamma }_k\left(y\right)=k^{\frac{y}{k}-1}\Gamma \left(\frac{y}{k}\right).$ Additionally, ${\Gamma }_k(y+k)=y{\Gamma }_k\left(y\right)$.

The following expression represents the complete beta function as defined in reference [40].

Definition 3.

The complete beta function is defined as

$$\begin{array}{c}\hfill \beta ({𝚤}_1,{𝚤}_2)={\int }_0^1{\xi }^{{𝚤}_1-1}{(1-\xi )}^{{𝚤}_2-1}d\xi ,\end{array}$$

where $Re\left({𝚤}_1\right)>0,Re\left({𝚤}_2\right)>0$.

The incomplete beta function [41] and relation between beta and gamma functions are given in the following.

Definition 4.

The incomplete beta function is defined as

$$\begin{array}{c}\hfill {\beta }_y({𝚤}_1,{𝚤}_2)={\int }_0^y{\xi }^{{𝚤}_1-1}{(1-\xi )}^{{𝚤}_2-1}d\xi ,\end{array}$$

where $y\in [0,1]$ and ${𝚤}_1>0,{𝚤}_2>0$. The notable relation between the beta and gamma function is stated as

$$\begin{array}{c}\hfill \beta ({𝚤}_1,{𝚤}_2)=\beta ({𝚤}_2,{𝚤}_1)=\frac{\Gamma \left({𝚤}_1\right)\left({𝚤}_2\right)}{\Gamma ({𝚤}_1+{𝚤}_2)}.\end{array}$$

The definition of the h-convex function presented in [42] is defined as follows.

Definition 5.

Let $h:\Omega \to \Re$ be a positive increasing function. A function $\stackrel{=}{\wedge }:\aleph \to \Re$ is h-convex, if $\stackrel{=}{\wedge }$ is nonnegative on ℵ, $\xi \in (0,1)$, and for all ${𝚤}_1,{𝚤}_2\in \aleph$, we have

$$\begin{array}{c}\hfill \stackrel{=}{\wedge }(\xi {𝚤}_1+(1-\xi ){𝚤}_2)\le h\left(\xi \right)\stackrel{=}{\wedge }\left({𝚤}_1\right)+h(1-\xi )\stackrel{=}{\wedge }\left({𝚤}_2\right).\end{array}$$

(1)

This definition generalizes the following convexities.

(i)

If we set $h\left(\xi \right)=l$ in (1), then we obtain the convex function

$$\begin{array}{c}\hfill \stackrel{=}{\wedge }(\xi {𝚤}_1+(1-\xi ){𝚤}_2)\le \xi \stackrel{=}{\wedge }\left({𝚤}_1\right)+(1-\xi )\stackrel{=}{\wedge }\left({𝚤}_2\right).\end{array}$$

(ii)

If we set $h\left(\xi \right)=l^s$ in (1), then we obtain the s-convex function in the second sense

$$\begin{array}{c}\hfill \stackrel{=}{\wedge }(\xi {𝚤}_1+(1-\xi ){𝚤}_2)\le {\xi }^s\stackrel{=}{\wedge }\left({𝚤}_1\right)+{(1-\xi )}^s\stackrel{=}{\wedge }\left({𝚤}_2\right).\end{array}$$

(iii)

If we set $h\left(\xi \right)=\frac{1}{\xi }$ in (1), then we obtain the Godunova Levin functions.

$$\begin{array}{c}\hfill \stackrel{=}{\wedge }(\xi {𝚤}_1+(1-\xi ){𝚤}_2)\le \frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)}{\xi }+\frac{\stackrel{=}{\wedge }\left({𝚤}_2\right)}{1-\xi }.\end{array}$$

Definition 6

([43]). Let $\stackrel{=}{\wedge }\in L[{𝚤}_1,{𝚤}_2]$, then the Riemann–Liouville fractional integrals of the order ζ are defined by

$$\begin{array}{c}\hfill \left({\mathcal{F}}_{{𝚤}_1^{+}}^{\zeta }\stackrel{=}{\wedge }\right)\left(y\right)=\frac{1}{\Gamma \left(\zeta \right)}{\int }_{{𝚤}_1}^y{(y-\xi )}^{\zeta -1}\stackrel{=}{\wedge }\left(\xi \right)d\xi ,(0\le {𝚤}_1<y),\end{array}$$

and

$$\begin{array}{c}\hfill \left({\mathcal{F}}_{{𝚤}_2^{-}}^{\zeta }\stackrel{=}{\wedge }\right)\left(y\right)=\frac{1}{\Gamma \left(\zeta \right)}{\int }_y^{{𝚤}_2}{(\xi -y)}^{\zeta -1}\stackrel{=}{\wedge }\left(\xi \right)d\xi ,(0\le y<{𝚤}_2),\end{array}$$

are known as the left- and right-sided Riemann–Liouville fractional integrals with $\Gamma (.)$ as the gamma function.

Definition 7

([39]). Let $\stackrel{=}{\wedge }\in L[{𝚤}_1,{𝚤}_2]$, then the fractional integrals of order ζ defined by

$$\begin{array}{c}\hfill \left({\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\right)\left(y\right)=\frac{1}{k{\Gamma }_k\left(\zeta \right)}{\int }_{{𝚤}_1}^y{(y-\xi )}^{\frac{\zeta }{k}-1}\stackrel{=}{\wedge }\left(\xi \right)d\xi ,(0\le {𝚤}_1<y),\end{array}$$

and

$$\begin{array}{c}\hfill \left({\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\right)\left(y\right)=\frac{1}{k{\Gamma }_k\left(\zeta \right)}{\int }_y^{{𝚤}_2}{(\xi -y)}^{\frac{\zeta }{k}-1}\stackrel{=}{\wedge }\left(\xi \right)d\xi ,(0\le y<{𝚤}_2),\end{array}$$

are known as the left- and right-sided k-Riemann–Liouville fractional integrals with a k-gamma function.

To establish the main results, we need to recall the following lemmas presented in [44,45], respectively.

Lemma 1.

For all $\xi \in [0,1]$, we have

$$\begin{array}{cc}\hfill & {(1-\xi )}^q\le 2^{1-q}-{\xi }^{q}forq\in [0,1],\hfill \\ \hfill & {(1-\xi )}^q\le 2^{1-q}-{\xi }^{q}forq\in [1,\infty ).\hfill \end{array}$$

Lemma 2.

Let $\stackrel{=}{\wedge }:[{𝚤}_1,{𝚤}_2]\to \Re$ be a function such that ${\stackrel{=}{\wedge }}^{″}$ exists on $({𝚤}_1,{𝚤}_2)$ with ${𝚤}_1<{𝚤}_2$ if ${\stackrel{=}{\wedge }}^{″}\in L[{𝚤}_1,{𝚤}_2]$; then, the equation for k-fractional integrals is as follows:

$$\begin{array}{cc}\hfill & \frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{2}-\frac{{\Gamma }_k(\zeta +k)}{2{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\hfill \\ \hfill & =\frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\int }_0^1\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1}{\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi .\hfill \end{array}$$

(2)

The motivation behind this work is to explore and analyze a new set of inequalities that involve mean-type concepts by employing more general convexity and fractional integrals. The Höder’s inequality is used to establish these results that have applications in diverse areas, including mathematics, statistics, engineering and computer science, where it serves as a valuable tool for analyzing and solving a wide range of problems. Through visual representations and validation of our findings, we seek to contribute to the existing body of knowledge and inspire further research in this area, potentially leading to advancements in mathematical theory and applications.

2. Main Results

In this section, we first present some important identities using a generalized fractional operator of the Riemann-type. Secondly, we explore Hermite–Hadamard inequalities by first establishing some identities.

Lemma 3.

Let $\stackrel{=}{\wedge }:[{𝚤}_1,{𝚤}_2]\to \Re$ be a function such that ${\stackrel{=}{\wedge }}^{″}$ exists on $({𝚤}_1,{𝚤}_2)$ with ${𝚤}_1<{𝚤}_2$ if ${\stackrel{=}{\wedge }}^{″}\in L[{𝚤}_1,{𝚤}_2]$; then, we have the identity

$$\begin{array}{cc}\hfill & \frac{{\Gamma }_k(\zeta +k)}{{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]-\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)\hfill \\ \hfill & ={({𝚤}_2-{𝚤}_1)}^2{\int }_0^{1}p\left(\xi \right){\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi ,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill p\left(\xi \right)=\left\{\begin{array}{cc}\xi -\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1},\hfill & \xi \in [0,\frac{1}{2}),\hfill \\ 1-\xi -\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1},\hfill & \xi \in [\frac{1}{2},1).\hfill \end{array}\right.\end{array}$$

(3)

Proof. 

Consider

$$\begin{array}{cc}\hfill & {\int }_0^{1}p\left(\xi \right){\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi \hfill \\ \hfill & ={\int }_0^{\frac{1}{2}}\left[\xi -\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1}\right]{\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi \hfill \\ \hfill & +{\int }_{\frac{1}{2}}^1\left[1-\xi -\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1}\right]{\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi \hfill \\ \hfill & ={\int }_0^{\frac{1}{2}}\xi {\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi +{\int }_{\frac{1}{2}}^1(1-\xi ){\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi \hfill \\ \hfill & -{\int }_0^1\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1}{\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi .\hfill \end{array}$$

(4)

Integrating by parts, we obtain

$$\begin{array}{cc}\hfill & {\int }_0^{\frac{1}{2}}\xi {\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi \hfill \\ \hfill & =\frac{1}{2({𝚤}_1-{𝚤}_2)}{\stackrel{=}{\wedge }}^{\prime }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)-\frac{1}{{({𝚤}_1-{𝚤}_2)}^2}\left[\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)-\stackrel{=}{\wedge }\left({𝚤}_2\right)\right]d\xi ,\hfill \end{array}$$

(5)

and

$$\begin{array}{cc}\hfill & {\int }_{\frac{1}{2}}^1\left(1-\xi \right){\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi \hfill \\ \hfill & =\frac{-1}{2({𝚤}_1-{𝚤}_2)}{\stackrel{=}{\wedge }}^{\prime }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)+\frac{1}{{({𝚤}_1-{𝚤}_2)}^2}\left[\stackrel{=}{\wedge }\left({𝚤}_1\right)-\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)\right].\hfill \end{array}$$

(6)

Substituting (5) and (6) in (4), we obtain

$$\begin{array}{cc}\hfill & {\int }_0^{1}p\left(\xi \right){\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi \hfill \\ \hfill & =\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{{({𝚤}_1-{𝚤}_2)}^2}-\frac{2}{{({𝚤}_1-{𝚤}_2)}^2}\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)\hfill \\ \hfill & -{\int }_0^1\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1}{\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi .\hfill \end{array}$$

(7)

Multiplying $\frac{{({𝚤}_2-{𝚤}_1)}^2}{2}$ with (7) on both sides, we can write

$$\begin{array}{cc}\hfill & \frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\int }_0^{1}p\left(\xi \right){\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi \hfill \\ \hfill & =\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{2}-\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)\hfill \\ \hfill & -\frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\int }_0^1\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1}{\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi .\hfill \end{array}$$

(8)

Substituting Lemma 2 into (8), we obtain the required result.

Hence, the proof is complete. □

Lemma 4.

Let $\stackrel{=}{\wedge }:[{𝚤}_1,{𝚤}_2]\to \Re$ be a function such that ${\stackrel{=}{\wedge }}^{″}$ exists on $({𝚤}_1,{𝚤}_2)$ with ${𝚤}_1<{𝚤}_2$ if ${\stackrel{=}{\wedge }}^{″}\in L[{𝚤}_1,{𝚤}_2]$, $⋋>0;$ then, the identity

$$\begin{array}{cc}\hfill & \frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{(⋋+1)}+\frac{2}{⋋+1}\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)-\frac{{\Gamma }_k(\zeta +k)}{⋋{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\hfill \\ \hfill & ={({𝚤}_2-{𝚤}_1)}^2{\int }_0^{1}m(l){\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi ,\hfill \end{array}$$

holds with

$$\begin{array}{c}\hfill m\left(\xi \right)=\left\{\begin{array}{cc}\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{⋋(\frac{\zeta }{k}+1)}-\frac{\xi }{⋋+1},\hfill & \xi \in [0,\frac{1}{2}),\hfill \\ \frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{⋋(\frac{\zeta }{k}+1)}-\frac{1-\xi }{⋋+1},\hfill & \xi \in [\frac{1}{2},1).\hfill \end{array}\right.\end{array}$$

(9)

Proof. 

Multiplying $\frac{1}{⋋(⋋+1)}$ with Lemma 2, then

$$\begin{array}{cc}\hfill & \frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{⋋(⋋+1)}-\frac{{\Gamma }_k(\zeta +k)}{⋋(⋋+1){({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\hfill \\ \hfill & =\frac{{({𝚤}_2-{𝚤}_1)}^2}{⋋(⋋+1)}{\int }_0^1\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1}{\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi .\hfill \end{array}$$

(10)

Multiplying $\frac{-1}{⋋+1}$ with Lemma 3, then

$$\begin{array}{cc}\hfill & \frac{2}{⋋+1}\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)-\frac{{\Gamma }_k(\zeta +k)}{(⋋+1){({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\hfill \\ \hfill & =-\frac{{({𝚤}_2-{𝚤}_1)}^2}{⋋+1}{\int }_0^{1}p\left(\xi \right){\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi .\hfill \end{array}$$

(11)

Hence, (10) and (11) yield

$$\begin{array}{cc}\hfill & \frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{⋋(⋋+1)}+\frac{2}{⋋+1}\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)-\frac{{\Gamma }_k(\zeta +k)}{⋋{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\hfill \\ \hfill & =\frac{{({𝚤}_2-{𝚤}_1)}^2}{⋋(⋋+1)}{\int }_0^1\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1}{\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi \hfill \\ \hfill & -\frac{{({𝚤}_2-{𝚤}_1)}^2}{⋋+1}{\int }_0^{1}p\left(\xi \right){\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi \hfill \\ \hfill & ={({𝚤}_2-{𝚤}_1)}^2{\int }_0^{1}m\left(\xi \right){\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)d\xi ,\hfill \end{array}$$

where $m\left(\xi \right)$ is defined by (9). The required proof is complete. □

By using Lemma 2 and Lemma 1, we obtain the following results.

Theorem 1.

Let $\stackrel{=}{\wedge }:[{𝚤}_1,{𝚤}_2]\to \Re$ be a function such that $|{\stackrel{=}{\wedge }}^{″}|$ exists and is measurable. Let $|{\stackrel{=}{\wedge }}^{″}|$ be a monotonic positive function and h-convex on $[{𝚤}_1,{𝚤}_2]$; then, for some fixed $\zeta \in (0,\infty )$, $0\le {𝚤}_1<{𝚤}_2$ the inequality

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{2}-\frac{{\Gamma }_k(\zeta +k)}{2{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\right|\hfill \\ \hfill & \le M\frac{k{({𝚤}_2-{𝚤}_1)}^2}{2(\zeta +k)}\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)\left(\frac{\zeta }{\zeta +2k}\right)\hfill \end{array}$$

(12)

holds, where h is bounded by M.

Proof. 

By using Lemma 2, we have

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{2}-\frac{{\Gamma }_k(\zeta +k)}{2{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\right|\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\int }_0^1\left|\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1}\right|\left|{\stackrel{=}{\wedge }}^{″}(\xi {𝚤}_1+(1-\xi ){𝚤}_2)\right|d\xi .\hfill \end{array}$$

By using the h-convexity of $|{\stackrel{=}{\wedge }}^{″}|$, we can write

$$\begin{array}{cc}\hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\int }_0^1\left|\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1}\right|\left(h\left(\xi \right)\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+h(1-\xi )\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)d\xi \hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\int }_0^1\left|\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1}\right|\left(M\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+M\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)d\xi \hfill \\ \hfill & \le M\frac{k{({𝚤}_2-{𝚤}_1)}^2}{2(\zeta +k)}\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)\left(\frac{\zeta }{\zeta +2k}\right).\hfill \end{array}$$

The required proof is complete. □

Example 1.

The inequality presented in Theorem 1 can be verified by sketching the graph of (29). For this purpose, we substitute $\stackrel{=}{\wedge }\left(\xi \right)=e^{\xi }$ and obtain the following

$$\begin{array}{c}\hfill {\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }{e}^{{𝚤}_2}=\frac{1}{k{\Gamma }_k\left(\zeta \right)}{\int }_{{𝚤}_1}^{{𝚤}_2}{({𝚤}_2-\xi )}^{\frac{\zeta }{k}-1}{e}^{\xi }d\xi ,\end{array}$$

(13)

and

$$\begin{array}{c}\hfill {\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }{e}^{{𝚤}_1}=\frac{1}{k{\Gamma }_k\left(\zeta \right)}{\int }_{{𝚤}_1}^{{𝚤}_2}{(\xi -{𝚤}_1)}^{\frac{\zeta }{k}-1}{e}^{\xi }d\xi .\end{array}$$

(14)

By utilizing the expressions (13) and (14) in (29), we obtain

$$\begin{array}{cc}\hfill & \frac{e^{{𝚤}_1}+e^{{𝚤}_2}}{2}-\frac{Mk{({𝚤}_2-{𝚤}_1)}^2}{2(\zeta +k)}\left(|e^{{𝚤}_1}|+|e^{{𝚤}_2}|\right)\left(\frac{\zeta }{\zeta +2k}\right)\hfill \\ \hfill & \le \frac{{\Gamma }_k(\zeta +k)}{2k{\Gamma }_k\left(\zeta \right){({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}{\int }_{{𝚤}_1}^{{𝚤}_2}[{({𝚤}_2-\xi )}^{\frac{\zeta }{k}-1}+{(\xi -{𝚤}_1)}^{\frac{\zeta }{k}-1}]e^{\xi }d\xi \hfill \\ \hfill & \le \frac{e^{{𝚤}_1}+e^{{𝚤}_2}}{2}+\frac{Mk{({𝚤}_2-{𝚤}_1)}^2}{2(\zeta +k)}\left(|e^{{𝚤}_1}|+|e^{{𝚤}_2}|\right)\left(\frac{\zeta }{\zeta +2k}\right).\hfill \end{array}$$

(15)

Corresponding to the choice of the parameters $M=1,{𝚤}_1=0$, ${𝚤}_2=1,$ $k=1$, with $1\le \zeta \le 4$, the graph of the double inequality (15) is as below.

$$\begin{array}{cc}\hfill & p_0\left(\zeta \right)=1.8591\left(1-\frac{1}{\zeta +1}\right)+\frac{3.7183}{(\zeta +1)(\zeta +2)}.\hfill \\ \hfill & p_1\left(\zeta \right)=\frac{\zeta }{2}{\int }_0^1\left[{(1-\xi )}^{\zeta -1}+{\xi }^{\zeta -1}\right]e^{\xi }d\xi .\hfill \\ \hfill & p_2\left(\zeta \right)=1.8591\left(1+\frac{1}{\zeta +1}\right)-\frac{3.7183}{(\zeta +1)(\zeta +2)}.\hfill \end{array}$$

The numerical values in

Table 1. The comparison results in Example 1 between the double inequality are presented in the following table.

Functions	1	1.5	2	2.5	3	3.5	4
$p_0\left(\zeta \right)$	$1.54927$	$1.54041$	$1.54926$	$1.56401$	$1.58024$	$1.5962$	$1.61122$
$p_1\left(\zeta \right)$	$1.71828$	$1.71428$	$1.71828$	$1.72494$	$1.73227$	$1.73947$	$1.74625$
$p_2\left(\zeta \right)$	$2.16893$	$2.17779$	$2.16894$	$2.15419$	$2.13796$	$2.122$	$2.10698$

corresponds to Figure 1.

Remark 1.

By substituting $h\left(\xi \right)=\xi$ and $k=1$ in Theorem 1, we arrive at ([46] Theorem 3.1) with a choice $s=1$, i.e.,

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{2}-\frac{\Gamma (\zeta +1)}{2{({𝚤}_2-{𝚤}_1)}^{\zeta }}\left[{\mathcal{F}}_{{𝚤}_1^{+}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\right|\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\zeta +1)}\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)\left(\frac{1}{2}-\frac{1}{\zeta +3}\right).\hfill \end{array}$$

Remark 2.

By substituting $h\left(\xi \right)={\xi }^s$ and $k=1$ in Theorem 1, we obtain ([46] Theorem 3.1), i.e.,

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{2}-\frac{\Gamma (\zeta +1)}{2{({𝚤}_2-{𝚤}_1)}^{\zeta }}\left[{\mathcal{F}}_{{𝚤}_1^{+}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\right|\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\zeta +1)}\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)\left(\frac{1}{s+1}-\frac{1}{\zeta +s+2}\right).\hfill \end{array}$$

Theorem 2.

Let the $\stackrel{=}{\wedge }:[{𝚤}_1,{𝚤}_2]\to \Re$ be a function such that the ${\stackrel{=}{\wedge }}^{″}$ exists. Let $0<{\omega }_2<\infty$, ${\left|{\stackrel{=}{\wedge }}^{″}\right|}^{{\omega }_2}\in L[{𝚤}_1,{𝚤}_2]$ be an h-convex monotonic positive function on $[{𝚤}_1,{𝚤}_2]$, $0\le {𝚤}_1<{𝚤}_2$. Then, the inequality

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{2}-\frac{{\Gamma }_k(\zeta +k)}{2{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\right|\hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^{2}max\left(1-2^{1-\frac{\zeta }{k}},2^{1-\frac{\zeta }{k}}-1\right)}{2(\frac{\zeta }{k}+1)}{\left({\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)}^{\frac{1}{{\omega }_2}}\hfill \end{array}$$

(16)

holds with $\left|h\right(x\left)\right|\le M$ and $\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}=1.$

Proof. 

The proof of this result is divided into two cases.

Case (i): Let $\zeta \in (0,1)$. By using Lemmas 1 and 2 and applying Hölder’s inequality,

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{2}-\frac{{\Gamma }_k(\zeta +k)}{2{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\right|\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\int }_0^1\left|\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1}\right|\left|{\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)\right|d\xi \hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}{\left({\int }_0^1{\left|1-{(1-\xi )}^{\frac{\zeta }{k}}-{\xi }^{\frac{\zeta }{k}}\right|}^{{\omega }_1}dl\right)}^{\frac{1}{{\omega }_1}}{\left({\int }_0^1{\left|{\stackrel{=}{\wedge }}^{″}(\xi {𝚤}_1+(1-\xi ){𝚤}_2)\right|}^{{\omega }_2}d\xi \right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}{\left({\int }_0^1{\left|{(1-\xi )}^{\frac{\zeta }{k}}+{\xi }^{\frac{\zeta }{k}}-1\right|}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}{\left({\int }_0^1{\left|{\stackrel{=}{\wedge }}^{″}(\xi {𝚤}_1+(1-\xi ){𝚤}_2)\right|}^{{\omega }_2}d\xi \right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}{\left({\int }_0^1{\left(2^{1-\frac{\zeta }{k}}-1\right)}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}{\left({\int }_0^1{\left|{\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)\right|}^{{\omega }_2}d\xi \right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}\left(2^{1-\frac{\zeta }{k}}-1\right){\left({\int }_0^1{\left|{\stackrel{=}{\wedge }}^{″}\left(\xi {𝚤}_1+(1-\xi ){𝚤}_2\right)\right|}^{{\omega }_2}d\xi \right)}^{\frac{1}{{\omega }_2}}.\hfill \end{array}$$

By using the h-convexity of $|{\stackrel{=}{\wedge }}^{″}{|}^{{\omega }_2}$,

$$\begin{array}{cc}\hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}\left(2^{1-\frac{\zeta }{k}}-1\right){\left({\int }_0^1\left(h\left(\xi \right){\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+h(1-\xi ){\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)d\xi \right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}\left(2^{1-\frac{\zeta }{k}}-1\right){\left({\int }_0^1\left(M{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+M{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)d\xi \right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}\left(2^{1-\frac{\zeta }{k}}-1\right){\left({\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)}^{\frac{1}{{\omega }_2}}.\hfill \end{array}$$

Case (ii): Let $\zeta \in [1,\infty )$. By using Lemmas 1 and 2 and applying Hölder’s inequality,

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{2}-\frac{{\Gamma }_k(\zeta +k)}{2{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\right|\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\int }_0^1\left|\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1}\right|\left|{\stackrel{=}{\wedge }}^{″}(\xi {𝚤}_1+(1-\xi ){𝚤}_2)\right|d\xi \hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}{\left({\int }_0^1{\left|1-{(1-\xi )}^{\frac{\zeta }{k}}-{\xi }^{\frac{\zeta }{k}}\right|}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}{\left({\int }_0^1{\left|{\stackrel{=}{\wedge }}^{″}(\xi {𝚤}_1+(1-\xi ){𝚤}_2)\right|}^{{\omega }_2}d\xi \right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}{\left({\int }_0^1{\left(1-2^{1-\frac{\zeta }{k}}\right)}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}{\left({\int }_0^1{\left|{\stackrel{=}{\wedge }}^{″}(\xi {𝚤}_1+(1-\xi ){𝚤}_2)\right|}^{{\omega }_2}d\xi \right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}\left(1-2^{1-\frac{\zeta }{k}}\right){\left({\int }_0^1{\left|{\stackrel{=}{\wedge }}^{″}(\xi {𝚤}_1+(1-\xi ){𝚤}_2)\right|}^{{\omega }_2}d\xi \right)}^{\frac{1}{{\omega }_2}}.\hfill \end{array}$$

By using the h-convexity of $|{\stackrel{=}{\wedge }}^{″}{|}^{{\omega }_2}$, we obtain

$$\begin{array}{cc}\hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}\left(1-2^{1-\frac{\zeta }{k}}\right){\left({\int }_0^1\left(h\left(\xi \right){\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+h(1-\xi ){\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)d\xi \right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}\left(1-2^{1-\frac{\zeta }{k}}\right){\left({\int }_0^1\left(M{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+M{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)d\xi \right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}\left(1-2^{1-\frac{\zeta }{k}}\right){\left({\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)}^{\frac{1}{{\omega }_2}}.\hfill \end{array}$$

Hence, the proof is complete. □

Example 2.

The inequality presented in Theorem 2 can be verified by sketching the graph of (31). For this purpose, we substitute $\stackrel{=}{\wedge }\left(\xi \right)={\xi }^2$ and obtain the following relations

$$\begin{array}{c}\hfill {\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }{𝚤}_2^2=\frac{1}{k{\Gamma }_k\left(\zeta \right)}{\int }_{{𝚤}_1}^{{𝚤}_2}{({𝚤}_2-\xi )}^{\frac{\zeta }{k}-1}{\xi }^{2}d\xi ,\end{array}$$

(17)

and

$$\begin{array}{c}\hfill {\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }{𝚤}_1^2=\frac{1}{k{\Gamma }_k\left(\zeta \right)}{\int }_{{𝚤}_1}^{{𝚤}_2}{(\xi -{𝚤}_1)}^{\frac{\zeta }{k}-1}{\xi }^{2}d\xi .\end{array}$$

(18)

By utilizing (17) and (18) in (31), we obtain

$$\begin{array}{cc}\hfill & \frac{{𝚤}_1^2+{𝚤}_2^2}{2}-\frac{M{({𝚤}_2-{𝚤}_1)}^{2}max(1-2^{1-\frac{\zeta }{k}},2^{1-\frac{\zeta }{k}}-1)}{2(\frac{\zeta }{k}+1)}{\left({\left|2\right|}^{{\omega }_2+1}\right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \le \frac{{\Gamma }_k(\zeta +k)}{2k{\Gamma }_k\left(\zeta \right){({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}{\int }_{{𝚤}_1}^{{𝚤}_2}[{({𝚤}_2-\xi )}^{\frac{\zeta }{k}-1}+{(\xi -{𝚤}_1)}^{\frac{\zeta }{k}-1}]{\xi }^{2}d\xi \hfill \\ \hfill & \le \frac{{𝚤}_1^2+{𝚤}_2^2}{2}+\frac{M{({𝚤}_2-{𝚤}_1)}^{2}max(1-2^{1-\frac{\zeta }{k}},2^{1-\frac{\zeta }{k}}-1)}{2(\frac{\zeta }{k}+1)}{\left({\left|2\right|}^{{\omega }_2+1}\right)}^{\frac{1}{{\omega }_2}}.\hfill \end{array}$$

(19)

Corresponding to the choice of the parameters $M=1,{𝚤}_1=0$, ${𝚤}_2=1$, ${\omega }_1=2$, ${\omega }_2=2$, $k=1$, with $0\le \zeta \le 2$, the graph of the double inequality (19) is as below.

$$\begin{array}{cc}\hfill & p_0\left(\zeta \right)=\frac{1}{2}-\frac{\sqrt{2}}{\zeta +1}.\hfill \\ \hfill & p_1\left(\zeta \right)=\frac{\zeta }{2}{\int }_0^1[{(1-\xi )}^{\zeta -1}+{\xi }^{\zeta -1}]{\xi }^{2}d\xi .\hfill \\ \hfill & p_2\left(\zeta \right)=\frac{1}{2}+\frac{\sqrt{2}}{\zeta +1}.\hfill \end{array}$$

The numerical values in

Table 2. The comparison results in Example 2 between the double inequality are presented in the following table.

Functions	0	0.4	0.8	1.2	1.6	2
$p_0\left(\zeta \right)$	$-0.914214$	$-0.510153$	$-0.285674$	$-0.142824$	$-0.0439283$	$0.0285955$
$p_1\left(\zeta \right)$	0	$0.380952$	$0.34127$	$0.329545$	$0.32906$	$0.333333$
$p_2\left(\zeta \right)$	$1.91421$	$1.51015$	$1.28567$	$1.14282$	$1.04393$	$0.971405$

corresponds to Figure 2.

Remark 3.

By substituting $h\left(\xi \right)=\xi$ and $k=1$ in Theorem 2, we obtain ([46] Theorem 3.2) with the choice $s=1$, i.e.,

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{2}-\frac{\Gamma (\zeta +1)}{2{({𝚤}_2-{𝚤}_1)}^{\zeta }}\left[{\mathcal{F}}_{{𝚤}_1^{+}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\right|\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^{2}max\left(1-2^{1-\zeta },2^{1-\zeta }-1\right)}{2(\zeta +1)}{\left(\frac{{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}}{2}\right)}^{\frac{1}{{\omega }_2}},\hfill \end{array}$$

where $\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}=1.$

Remark 4.

By substituting $h\left(\xi \right)={\xi }^s$ and $k=1$ in Theorem 2, we obtain ([46] Theorem 3.2), i.e.,

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{2}-\frac{\Gamma (\zeta +1)}{2{({𝚤}_2-{𝚤}_1)}^{\zeta }}\left[{\mathcal{F}}_{{𝚤}_1^{+}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\right|\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^{2}max\left(1-2^{1-\zeta },2^{1-\zeta }-1\right)}{2(\zeta +1)}{\left(\frac{{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}}{s+1}\right)}^{\frac{1}{{\omega }_2}},\hfill \end{array}$$

where $\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}=1.$

In the next two results, we used Lemma 3.

Theorem 3.

Let the $\stackrel{=}{\wedge }:[{𝚤}_1,{𝚤}_2]\to \Re$ be a differentiable mapping $|{\stackrel{=}{\wedge }}^{″}|$, is measurable, and $|{\stackrel{=}{\wedge }}^{″}|$ is a monotonic positive function and h-convex on $[{𝚤}_1,{𝚤}_2]$, $0\le {𝚤}_1<{𝚤}_2$; then, we have the result

$$\begin{array}{cc}\hfill & \left|\frac{{\Gamma }_k(\zeta +k)}{2{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]-\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)\right|\hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{2}\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)\left(\frac{1}{4}-\frac{1}{\frac{\zeta }{k}+1}+\frac{2}{(\frac{\zeta }{k}+1)(\frac{\zeta }{k}+2)}\right),\hfill \end{array}$$

(20)

where $\left|h\right(x\left)\right|\le M$ and $\zeta \in (0,\infty )$.

Proof. 

By using Lemma 3, we can write

$$\begin{array}{cc}\hfill & \left|\frac{{\Gamma }_k(\zeta +k)}{2{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]-\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)\right|\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\int }_0^1\left|p\left(\xi \right)\right|\left|{\stackrel{=}{\wedge }}^{″}(\xi {𝚤}_1+(1-\xi ){𝚤}_2)\right|d\xi .\hfill \end{array}$$

By using the h-convexity of $|{\stackrel{=}{\wedge }}^{″}|$, we have

$$\begin{array}{cc}\hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\int }_0^1\left(\left|p\left(\xi \right)\right|\left(h\left(\xi \right)\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+h(1-\xi )\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)\right)d\xi \hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\int }_0^1\left|p\left(\xi \right)\right|\left(M\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+M\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)d\xi \hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\int }_0^1\left(\left|p\left(\xi \right)\right|\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)\right)d\xi \hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{2}\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right){\int }_0^1\left|p\left(\xi \right)\right|d\xi \hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{2}\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)\left({\int }_0^{\frac{1}{2}}\left|\xi -\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1}\right|d\xi \right.\hfill \\ \hfill & \left.+{\int }_{\frac{1}{2}}^1\left|1-\xi -\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1}\right|d\xi \right)\hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)\left({\int }_0^{\frac{1}{2}}\left|\xi \frac{\zeta }{k}+\xi -1+{(1-\xi )}^{\frac{\zeta }{k}+1}\right.\right.\hfill \\ \hfill & \left.\left.+{\xi }^{\frac{\zeta }{k}+1}\right|d\xi +{\int }_{\frac{1}{2}}^1\left|\frac{\zeta }{k}+1-\xi \frac{\zeta }{k}-1-\xi +{(1-\xi )}^{\frac{\zeta }{k}+1}+{\xi }^{\frac{\zeta }{k}+1}\right|d\xi \right)\hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{2}\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)\left(\frac{1}{4}-\frac{1}{\frac{\zeta }{k}+1}+\frac{2}{(\frac{\zeta }{k}+1)(\frac{\zeta }{k}+2)}\right).\hfill \end{array}$$

Hence, the desired result is proved. □

Example 3.

The inequality presented in Theorem 3 can be verified by sketching the graph of (20). For this purpose, by utilizing the expressions (13) and (14) in (20), we obtain

$$\begin{array}{cc}\hfill & e^{\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)}-\frac{M{({𝚤}_2-{𝚤}_1)}^2}{2}\left(\left|e^{{𝚤}_1}\right|+\left|e^{{𝚤}_2}\right|\right)\left(\frac{1}{4}-\frac{1}{\frac{\zeta }{k}+1}+\frac{2}{(\frac{\zeta }{k}+1)(\frac{\zeta }{k}+2)}\right)\hfill \\ \hfill & \le \frac{{\Gamma }_k(\zeta +k)}{2k{\Gamma }_k\left(\zeta \right){({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}{\int }_{{𝚤}_1}^{{𝚤}_2}\left[{({𝚤}_2-\xi )}^{\frac{\zeta }{k}-1}+{(\xi -{𝚤}_1)}^{\frac{\zeta }{k}-1}\right]e^{\xi }d\xi \hfill \\ \hfill & \le e^{\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)}+\frac{M{({𝚤}_2-{𝚤}_1)}^2}{2}\left(\left|e^{{𝚤}_1}\right|+\left|e^{{𝚤}_2}\right|\right)\left(\frac{1}{4}-\frac{1}{\frac{\zeta }{k}+1}+\frac{2}{(\frac{\zeta }{k}+1)(\frac{\zeta }{k}+2)}\right).\hfill \end{array}$$

(21)

Corresponding to the choice of the parameters $M=1,{𝚤}_1=0$, ${𝚤}_2=1$, $k=1$, with $1\le \zeta \le 4$, the graph of the double inequality (21) is as below.

$$\begin{array}{cc}\hfill & p_0\left(\zeta \right)=1.6487-1.8591\left(\frac{1}{4}-\frac{1}{\zeta +1}+\frac{2}{(\zeta +1)(\zeta +2)}\right).\hfill \\ \hfill & p_1\left(\zeta \right)=\frac{\zeta }{2}{\int }_0^1\left[{(1-\xi )}^{\zeta -1}+{\xi }^{\zeta -1}\right]e^{\xi }d\xi .\hfill \\ \hfill & p_2\left(\zeta \right)=1.6487+1.8591\left(\frac{1}{4}-\frac{1}{\zeta +1}+\frac{2}{(\zeta +1)(\zeta +2)}\right).\hfill \end{array}$$

The numerical values in

Table 3. The comparison results in Example 3 between the double inequality are presented in the following table.

Functions	1	1.5	2	2.5	3	3.5	4
$p_0\left(\zeta \right)$	$1.49378$	$1.50263$	$1.49378$	$1.47902$	$1.46279$	$1.44683$	$1.43181$
$p_1\left(\zeta \right)$	$1.71828$	$1.71428$	$1.71828$	$1.72494$	$1.73227$	$1.73947$	$1.74625$
$p_2\left(\zeta \right)$	$1.80363$	$1.79477$	$1.80363$	$1.81838$	$1.83461$	$1.85057$	$1.8656$

corresponds to Figure 3.

Remark 5.

By substituting $h\left(\xi \right)=\xi$ and $k=1$ in Theorem 3, we obtain ([46] Theorem 4.1) with the choice $s=1.$

$$\begin{array}{cc}\hfill & \left|\frac{\Gamma (\zeta +1)}{2{({𝚤}_2-{𝚤}_1)}^{\zeta }}\left[{\mathcal{F}}_{{𝚤}_1^{+}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]-\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)\right|\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}{2(\zeta +1)}\left[\frac{1}{\zeta +3}-\frac{\zeta +1}{8}+2\beta (2,\zeta +2)\right]\hfill \\ \hfill & +\frac{{({𝚤}_2-{𝚤}_1)}^2\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}{2(\zeta +1)}\left[\frac{1}{\zeta +3}-\frac{\zeta -5}{8}+2\beta (\zeta +2,2)\right].\hfill \end{array}$$

Remark 6.

By substituting $h\left(\xi \right)={\xi }^s$ and $k=1$ in Theorem 3, we obtain ([46] Theorem 4.1).

$$\begin{array}{cc}\hfill & \left|\frac{\Gamma (\zeta +1)}{2{({𝚤}_2-{𝚤}_1)}^{\zeta }}\left[{\mathcal{F}}_{{𝚤}_1^{+}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]-\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)\right|\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}{2(\zeta +1)}\left[\frac{\zeta -\zeta 2^{-s-1}-2^{-s-1}}{1+s}-\frac{\zeta +1}{2+s}\right.\hfill \\ \hfill & \left.+2\beta (\zeta +2,s+1)+\frac{1}{\zeta +s+2}\right]\hfill \\ \hfill & +\frac{{({𝚤}_2-{𝚤}_1)}^2\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}{2(\zeta +1)}\left[\frac{\zeta 2^{-s-1}-2^{-s-1}-1}{1+s}-\frac{1}{\zeta +s+2}+2\beta (\zeta +2,s+1)\right].\hfill \end{array}$$

Theorem 4.

Consider a function $\stackrel{=}{\wedge }$ defined on the interval $[{𝚤}_1,{𝚤}_2]$ such that its second derivative, denoted by ${\stackrel{=}{\wedge }}^{″}$, exists on this interval. Assume that the function ${\left|{\stackrel{=}{\wedge }}^{″}\right|}^{{\omega }_2}$ belongs to the class of integrable functions $L[{𝚤}_1,{𝚤}_2]$ and is both h-convex and a positive monotonic function. Additionally, suppose that $0\le {𝚤}_1<{𝚤}_2$. Then, the inequality

$$\begin{array}{cc}\hfill & \left|\frac{{\Gamma }_k(\zeta +k)}{2{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]-\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)\right|\hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}{\left({\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \times {\left(\frac{\left(\frac{\zeta }{k}+1\right)2^{-{\omega }_1-1}-{\left(\frac{\zeta }{k}\right)}^{{\omega }_1+1}+{\left(\frac{\zeta }{k}+0.5\right)}^{{\omega }_1+1}}{{\omega }_1+1}\right)}^{\frac{1}{{\omega }_1}}\hfill \end{array}$$

(22)

holds, where $\left|h\right(x\left)\right|\le M$ and $\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}=1$.

Proof. 

By using Hölder’s inequality and Lemma 3, we have

$$\begin{array}{c}\left|\frac{{\Gamma }_k(\zeta +k)}{2{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]-\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)\right|\hfill \\ \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\int }_0^1\left(\left|p\left(\xi \right)\right|\left|{\stackrel{=}{\wedge }}^{″}(\xi {𝚤}_1+(1-\xi ){𝚤}_2)\right|\right)d\xi \hfill \\ \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\left({\int }_0^1{\left|p\left(\xi \right)\right|}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}{\left({\int }_0^1{\left|{\stackrel{=}{\wedge }}^{″}(\xi {𝚤}_1+(1-\xi ){𝚤}_2)\right|}^{{\omega }_2}d\xi \right)}^{\frac{1}{{\omega }_2}}.\hfill \end{array}$$

By using the h-convexity of $|{\stackrel{=}{\wedge }}^{″}{|}^{{\omega }_2}$, we obtain

$$\begin{array}{cc}\hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\left({\int }_0^1{\left|p\left(\xi \right)\right|}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}\hfill \\ \hfill & \times {\left({\int }_0^1\left(h\left(\xi \right){\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+h(1-\xi ){\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)d\xi \right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\left({\int }_0^1{\left|p\left(\xi \right)\right|}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}{\left({\int }_0^1\left(M{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+M{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)d\xi \right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\left({\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)}^{\frac{1}{{\omega }_2}}{\left({\int }_0^1{\left|p\left(\xi \right)\right|}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}\hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{2}{\left({\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \times \left({\int }_0^{\frac{1}{2}}{\left|\xi -\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1}\right|}^{{\omega }_1}d\xi \right.\hfill \\ \hfill & {\left.+{\int }_{\frac{1}{2}}^1{\left|1-\xi -\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{\frac{\zeta }{k}+1}\right|}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}\hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}{\left({\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \times \left({\int }_0^{\frac{1}{2}}{\left|\xi \frac{\zeta }{k}+\xi -1+{(1-\xi )}^{\frac{\zeta }{k}+1}+{\xi }^{\frac{\zeta }{k}+1}\right|}^{{\omega }_1}d\xi \right.\hfill \\ \hfill & {\left.+{\int }_{\frac{1}{2}}^1{\left|\frac{\zeta }{k}+1-\xi \frac{\zeta }{k}-1-\xi +{(1-\xi )}^{\frac{\zeta }{k}+1}+{\xi }^{\frac{\zeta }{k}+1}\right|}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}\hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{2\left((\frac{\zeta }{k}+1)\right)}{\left({\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \times {\left({\int }_0^{\frac{1}{2}}{\left|(\frac{\zeta }{k}+1)\xi \right|}^{{\omega }_1}d\xi +{\int }_{\frac{1}{2}}^1{\left|\frac{\zeta }{k}-\xi +1\right|}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}\hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}{\left({\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \times {\left({\int }_0^{\frac{1}{2}}\left(\frac{\zeta }{k}+1\right){\xi }^{{\omega }_1}d\xi +{\int }_{\frac{1}{2}}^1{\left(\frac{\zeta }{k}-\xi +1\right)}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}\hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\frac{\zeta }{k}+1)}{\left({\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \times {\left(\frac{(\frac{\zeta }{k}+1)2^{-{\omega }_1-1}-{\left(\frac{\zeta }{k}\right)}^{{\omega }_1+1}+{(\frac{\zeta }{k}+0.5)}^{{\omega }_1+1}}{{\omega }_1+1}\right)}^{\frac{1}{{\omega }_1}}.\hfill \end{array}$$

This completes the result. □

Remark 7.

By substituting $h\left(\xi \right)=\xi$ and $k=1$ in Theorem 4, we obtain ([46] Theorem 4.2) with the choice $s=1.$

$$\begin{array}{cc}\hfill & \frac{\Gamma (\zeta +1)}{2{({𝚤}_2-{𝚤}_1)}^{\zeta }}\left[{\mathcal{F}}_{{𝚤}_1^{+}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]-\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\zeta +1)}{\left(\frac{{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}}{2}\right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \times {\left(\frac{(\zeta +1)2^{-{\omega }_1-1}+{(\zeta +0.5)}^{{\omega }_1+1}-{\zeta }^{{\omega }_1+1}}{{\omega }_1+1}\right)}^{\frac{1}{{\omega }_1}},\hfill \end{array}$$

where $\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}=1.$

Remark 8.

By substituting $h\left(\xi \right)={\xi }^s$ and $k=1$ in Theorem 4, we obtain ([46] Theorem 4.2).

$$\begin{array}{cc}\hfill & \frac{\Gamma (\zeta +1)}{2{({𝚤}_2-{𝚤}_1)}^{\zeta }}\left[{\mathcal{F}}_{{𝚤}_1^{+}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]-\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{2(\zeta +1)}{\left(\frac{{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}}{s+1}\right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \times {\left(\frac{(\zeta +1)2^{-{\omega }_1-1}+{(\zeta +0.5)}^{{\omega }_1+1}-{\zeta }^{{\omega }_1+1}}{{\omega }_1+1}\right)}^{\frac{1}{{\omega }_1}},\hfill \end{array}$$

where $\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}=1.$

By using Lemma 4, we obtain the next two results.

Theorem 5.

Suppose we have a function $\stackrel{=}{\wedge }$ defined on the interval $[{𝚤}_1,{𝚤}_2]$ such that the absolute value of its second derivative, denoted by $|{\stackrel{=}{\wedge }}^{″}|$, exists. If $|{\stackrel{=}{\wedge }}^{″}|$ is both monotonic and positive and it is also h-convex on the interval $[{𝚤}_1,{𝚤}_2]$, where $0\le {𝚤}_1<{𝚤}_2$, then the following inequality holds:

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{⋋(⋋+1)}+\frac{2}{⋋+1}\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)-\frac{{\Gamma }_k(\zeta +k)}{⋋{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\right|\hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{⋋(⋋+1)(\frac{\zeta }{k}+1)}\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)\hfill \\ \hfill & \times max\left[\frac{⋋+1}{2}-\frac{⋋(\frac{\zeta }{k}+1)}{8}-\frac{⋋+1}{\frac{\zeta }{k}+2},\frac{-⋋-1}{2}+\frac{⋋(\frac{\zeta }{k}+1)}{8}+\frac{⋋+1}{\frac{\zeta }{k}+2}\right]\hfill \\ \hfill & +M\frac{{({𝚤}_2-{𝚤}_1)}^2}{⋋(⋋+1)(\frac{\zeta }{k}+1)}\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)\hfill \\ \hfill & \times max\left[\frac{⋋+1}{2}-\frac{⋋(\frac{\zeta }{k}+1)}{8}-\frac{⋋+1}{\frac{\zeta }{k}+2},\frac{-⋋-1}{2}+\frac{⋋(\frac{\zeta }{k}+1)}{8}+\frac{⋋+1}{\frac{\zeta }{k}+2}\right],\hfill \end{array}$$

(23)

where $\left|h\right(x\left)\right|\le M.$

Proof. 

By using Lemma 4, we can write

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{⋋(⋋+1)}+\frac{2}{⋋+1}\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)-\frac{{\Gamma }_k(\zeta +k)}{⋋{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\right|\hfill \\ \hfill & \le {({𝚤}_2-{𝚤}_1)}^2{\int }_0^1\left(\left|m\left(\xi \right)\right|\left|{\stackrel{=}{\wedge }}^{″}(\xi {𝚤}_1+(1-\xi ){𝚤}_2)\right|\right)d\xi .\hfill \end{array}$$

By using the h-convexity of $|{\stackrel{=}{\wedge }}^{″}|$, we can write

$$\begin{array}{cc}\hfill & \le {({𝚤}_2-{𝚤}_1)}^2{\int }_0^1\left|m\left(\xi \right)\right|\left(h\left(\xi \right)\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+h(1-\xi )\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)d\xi \hfill \\ \hfill & \le {({𝚤}_2-{𝚤}_1)}^2{\int }_0^1\left|m\left(\xi \right)\right|\left(M\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+M\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)d\xi \hfill \\ \hfill & \le M{({𝚤}_2-{𝚤}_1)}^2\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right){\int }_0^1\left|m\left(\xi \right)\right|d\xi \hfill \\ \hfill & \le M{({𝚤}_2-{𝚤}_1)}^2\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)\hfill \\ \hfill & \times \left({\int }_0^{\frac{1}{2}}\left|\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{⋋(\frac{\zeta }{k}+1)}-\frac{\xi }{⋋+1}\right|d\xi \right.\hfill \\ \hfill & +\left.{\int }_{\frac{1}{2}}^1\left|\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{⋋(\frac{\zeta }{k}+1)}-\frac{1-\xi }{⋋+1}\right|d\xi \right)\hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{⋋(⋋+1)(\frac{\zeta }{k}+1)}\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)\hfill \\ \hfill & \times \left({\int }_0^{\frac{1}{2}}\left|⋋+1-(⋋+1)({(1-\xi )}^{\frac{\zeta }{k}+1}+{\xi }^{\frac{\zeta }{k}+1})-⋋(\frac{\zeta }{k}+1)\xi \right|d\xi \right.\hfill \\ \hfill & +\left.{\int }_{\frac{1}{2}}^1\left|⋋+1-(⋋+1)({(1-\xi )}^{\frac{\zeta }{k}+1}+{\xi }^{\frac{\zeta }{k}+1})-⋋(\frac{\zeta }{k}+1)(1-\xi )\right|d\xi \right).\hfill \end{array}$$

(24)

However,

$$\begin{array}{cc}\hfill & {\int }_0^{\frac{1}{2}}\left[⋋+1-(⋋+1)({(1-\xi )}^{\frac{\zeta }{k}+1}+{\xi }^{\frac{\zeta }{k}+1})-⋋(\frac{\zeta }{k}+1)\xi \right]d\xi \hfill \\ \hfill & \le \frac{⋋+1}{2}-\frac{⋋(\frac{\zeta }{k}+1)}{8}-\frac{⋋+1}{\frac{\zeta }{k}+2},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\int }_0^{\frac{1}{2}}\left[-⋋-1+(⋋+1)({(1-\xi )}^{\frac{\zeta }{k}+1}+{\xi }^{\frac{\zeta }{k}+1})+⋋(\frac{\zeta }{k}+1)\xi \right]d\xi \hfill \\ \hfill & \le \frac{-⋋-1}{2}+\frac{⋋(\frac{\zeta }{k}+1)}{8}+\frac{⋋+1}{\frac{\zeta }{k}+2}.\hfill \end{array}$$

Additionally,

$$\begin{array}{cc}\hfill & {\int }_{\frac{1}{2}}^1\left[⋋+1-(⋋+1)({(1-\xi )}^{\frac{\zeta }{k}+1}+{\xi }^{\frac{\zeta }{k}+1})-⋋(\frac{\zeta }{k}+1)(1-\xi )\right]d\xi \hfill \\ \hfill & \le \frac{⋋+1}{2}-\frac{⋋(\frac{\zeta }{k}+1)}{8}-\frac{⋋+1}{\frac{\zeta }{k}+2},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\int }_{\frac{1}{2}}^1\left[-⋋-1+(⋋+1)({(1-\xi )}^{\frac{\zeta }{k}+1}+{\xi }^{\frac{\zeta }{k}+1})+⋋(\frac{\zeta }{k}+1)(1-\xi )\right]d\xi \hfill \\ \hfill & \le \frac{-⋋-1}{2}+\frac{⋋(\frac{\zeta }{k}+1)}{8}+\frac{⋋+1}{\frac{\zeta }{k}+2}.\hfill \end{array}$$

Substituting these values of integrals in (24), we obtain the required result. □

Example 4.

The inequality presented in Theorem 5 can be verified by sketching the graph of (23). For this purpose, by utilizing (17) and (18) in (23), we obtain

$$\begin{array}{cc}\hfill & \frac{{𝚤}_1^2+{𝚤}_2^2}{⋋(⋋+1)}+\frac{2}{⋋+1}{\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)}^2-M\frac{4{({𝚤}_2-{𝚤}_1)}^2}{⋋(⋋+1)(\frac{\zeta }{k}+1)}\hfill \\ \hfill & \times max\left[\frac{⋋+1}{2}-\frac{⋋(\frac{\zeta }{k}+1)}{8}-\frac{⋋+1}{\frac{\zeta }{k}+2},\frac{-⋋-1}{2}+\frac{⋋(\frac{\zeta }{k}+1)}{8}+\frac{⋋+1}{\frac{\zeta }{k}+2}\right]\hfill \\ \hfill -& M\frac{4{({𝚤}_2-{𝚤}_1)}^2}{⋋(⋋+1)(\frac{\zeta }{k}+1)}\hfill \\ \hfill & \times max\left[\frac{⋋+1}{2}-\frac{⋋(\frac{\zeta }{k}+1)}{8}-\frac{⋋+1}{\frac{\zeta }{k}+2},\frac{-⋋-1}{2}+\frac{⋋(\frac{\zeta }{k}+1)}{8}+\frac{⋋+1}{\frac{\zeta }{k}+2}\right]\hfill \\ \hfill & \le \frac{{\Gamma }_k(\zeta +k)}{k⋋{\Gamma }_k\left(\zeta \right){({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}{\int }_{{𝚤}_1}^{{𝚤}_2}\left[{({𝚤}_2-\xi )}^{\frac{\zeta }{k}-1}+{(\xi -{𝚤}_1)}^{\frac{\zeta }{k}-1}\right]e^{\xi }d\xi \hfill \\ \hfill & \le \frac{{𝚤}_1^2+{𝚤}_2^2}{⋋(⋋+1)}+\frac{2}{⋋+1}{\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)}^2+M\frac{4{({𝚤}_2-{𝚤}_1)}^2}{⋋(⋋+1)(\frac{\zeta }{k}+1)}\hfill \\ \hfill & \times max\left[\frac{⋋+1}{2}-\frac{⋋(\frac{\zeta }{k}+1)}{8}-\frac{⋋+1}{\frac{\zeta }{k}+2},\frac{-⋋-1}{2}+\frac{⋋(\frac{\zeta }{k}+1)}{8}+\frac{⋋+1}{\frac{\zeta }{k}+2}\right]\hfill \\ \hfill +& M\frac{4{({𝚤}_2-{𝚤}_1)}^2}{⋋(⋋+1)(\frac{\zeta }{k}+1)}\hfill \\ \hfill & \times max\left[\frac{⋋+1}{2}-\frac{⋋(\frac{\zeta }{k}+1)}{8}-\frac{⋋+1}{\frac{\zeta }{k}+2},\frac{-⋋-1}{2}+\frac{⋋(\frac{\zeta }{k}+1)}{8}+\frac{⋋+1}{\frac{\zeta }{k}+2}\right].\hfill \end{array}$$

(25)

Corresponding to the choice of the parameters $M=1,{𝚤}_1=0$, ${𝚤}_2=1$, $⋋=1$, $k=1$, with $0\le \zeta \le 2$, the graph of the double inequality (25) is as below.

$$\begin{array}{cc}\hfill & p_0\left(\zeta \right)=\frac{3}{4}-\frac{1}{\zeta +1}.\hfill \\ \hfill & p_1\left(\zeta \right)=\zeta {\int }_0^1[{(1-\xi )}^{\zeta -1}+{\xi }^{\zeta -1}]{\xi }^{2}d\xi .\hfill \\ \hfill & p_2\left(\zeta \right)=\frac{3}{4}+\frac{1}{\zeta +1}.\hfill \end{array}$$

The numerical values in

Table 4. The comparison results in Example 4 between the double inequality are presented in the following table.

Functions	0	0.4	0.8	1.2	1.6	2
$p_0\left(\zeta \right)$	$-0.25$	$0.357143$	$0.194444$	$0.295455$	$0.365385$	$0.416667$
$p_1\left(\zeta \right)$	0	$0.761905$	$0.68254$	$0.659091$	$0.65812$	$0.666667$
$p_2\left(\zeta \right)$	$1.75$	$1.46429$	$1.30556$	$1.20455$	$1.13462$	$1.083333$

corresponds to Figure 4.

Remark 9.

By substituting $h\left(\xi \right)=\xi$ and $k=1$ in Theorem 5, we obtain ([46] Theorem 5.1) with the choice $s=1.$

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{⋋(⋋+1)}+\frac{2}{⋋+1}\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)-\frac{\Gamma (\zeta +1)}{⋋{({𝚤}_2-{𝚤}_1)}^{\zeta }}\left[{\mathcal{F}}_{{𝚤}_1^{+}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\right|\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{⋋(⋋+1)(\zeta +1)}max\left(\left[⋋+1-(⋋+1)2^{-\zeta }\right]\left[\frac{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+3\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}{8}\right]\right.\hfill \\ \hfill & \left.-⋋(\zeta +1)\left[\frac{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+57\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}{24}\right],⋋(\zeta +1)\left[\frac{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+57\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}{24}\right]\right)\hfill \\ \hfill & +\frac{{({𝚤}_2-{𝚤}_1)}^2}{⋋(⋋+1)(\zeta +1)}max\left(\left[1-⋋\zeta -(⋋+1)2^{-\zeta }\right]\left[\frac{3\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|-\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}{8}\right]\right.\hfill \\ \hfill & \left.+⋋(\zeta +1)\left[\frac{7\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+57\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}{24}\right],⋋(\zeta +1)\left[\frac{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+27\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}{12}\right]\right).\hfill \end{array}$$

Remark 10.

By substituting $h\left(\xi \right)={\xi }^s$ and $k=1$ in Theorem 5, we obtain ([46] Theorem 5.1).

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{⋋(⋋+1)}+\frac{2}{⋋+1}\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)-\frac{\Gamma (\zeta +1)}{⋋{({𝚤}_2-{𝚤}_1)}^{\zeta }}\left[{\mathcal{F}}_{{𝚤}_1^{+}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\right|\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{⋋(⋋+1)(\zeta +1)}max\left(\left[\frac{2^{-s-1}\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+(1-2^{-s-1})\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}{s+1}\right]\right.\hfill \\ \hfill & \times \left[⋋+1-(⋋+1)2^{-\zeta }\right]-⋋(\zeta +1)\left[\frac{2^{-s-2}\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}{s+2}+{\beta }_{0.5}(2,s+1)\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right]\hfill \\ \hfill & \left.,⋋(\zeta +1)\left[\frac{2^{-s-2}\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}{s+2}+{\beta }_{0.5}(2,s+1)\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right]\right)\hfill \\ \hfill & +\frac{{({𝚤}_2-{𝚤}_1)}^2}{⋋(⋋+1)(\zeta +1)}max\left(\left[\frac{(1-2^{-s-1})\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|-2^{-s-1}\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}{s+1}\right]\right.\hfill \\ \hfill & \times \left[⋋+1-(⋋+1)2^{-\zeta }-⋋(\zeta +1)\right]+⋋(\zeta +1)\left[\frac{(1-2^{-s-2})\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}{s+2}\right.\hfill \\ \hfill & \left.+{\beta }_{0.5}(s+1,2)\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right],\hfill \\ \hfill & ⋋(\zeta +1)\left[\frac{(1-2^{-s-1})\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|-2^{-s-1}\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}{s+1}-\frac{(1-2^{-s-2})\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}{s+2}\right.\hfill \\ \hfill & \left.\left.-{\beta }_{0.5}(s+1,2)\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right]\right).\hfill \end{array}$$

Theorem 6.

Let the $\stackrel{=}{\wedge }:[{𝚤}_1,{𝚤}_2]\to \Re$ be a function such that $|{\stackrel{=}{\wedge }}^{″}|$ exists on $[{𝚤}_1,{𝚤}_2].$ Let $0<{\omega }_2<\infty$, $|{\stackrel{=}{\wedge }}^{″}{|}^q\in L[{𝚤}_1,{𝚤}_2]$ be an h-convex, monotonic positive function where $0\le {𝚤}_1<{𝚤}_2$. Then, the following inequality

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{⋋(⋋+1)}+\frac{2}{⋋+1}\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)-\frac{{\Gamma }_k(\zeta +k)}{⋋{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\right|\hfill \\ \hfill & \le \frac{M{({𝚤}_2-{𝚤}_1)}^2}{{[⋋(⋋+1)(\frac{\zeta }{k}+1)]}^{1+{\omega }_1^{-1}}}{\left({\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \times \left(max\left[{\left[⋋+1-(⋋+1)2^{-\frac{\zeta }{k}}\right]}^{{\omega }_1+1}-{\left[1+0.5⋋(1-\frac{\zeta }{k})-(1+⋋)2^{-\frac{\zeta }{k}}\right]}^{{\omega }_1+1},\right.\right.\hfill \\ \hfill & \left.{\left[⋋(\frac{\zeta }{k}+1)\right]}^{{\omega }_1+1}{2}^{-{\omega }_1-1}\right]+max\left[{\left[⋋+1-(⋋+1)2^{-\frac{\zeta }{k}}\right]}^{{\omega }_1+1}\right.\hfill \\ \hfill & {\left.\left.-{\left[1+0.5⋋(1-\frac{\zeta }{k})-(⋋+1)2^{-\frac{\zeta }{k}}\right]}^{{\omega }_1+1},{\left[0.5⋋(\frac{\zeta }{k}+1)\right]}^{{\omega }_1+1}\right]\right)}^{\frac{1}{{\omega }_1}}\hfill \end{array}$$

(26)

holds, where $\left|h\right(x\left)\right|\le M$ and $\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}=1.$

Proof. 

By using the Hölder’s inequality and Lemma 4, we can write

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{⋋(⋋+1)}+\frac{2}{⋋+1}\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)-\frac{{\Gamma }_k(\zeta +k)}{⋋{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}\left[{\mathcal{F}}_{{𝚤}_1^{+},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-},k}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\right|\hfill \\ \hfill & \le {({𝚤}_2-{𝚤}_1)}^2{\int }_0^1\left(\left|m\left(\xi \right)\right|\left|{\stackrel{=}{\wedge }}^{″}(\xi {𝚤}_1+(1-\xi ){𝚤}_2)\right|\right)d\xi \hfill \\ \hfill & \le {({𝚤}_2-{𝚤}_1)}^2{\left({\int }_0^1{\left|m\left(\xi \right)\right|}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}{\left({\int }_0^1{\left|{\stackrel{=}{\wedge }}^{″}(\xi {𝚤}_1+(1-\xi ){𝚤}_2)\right|}^{{\omega }_2}d\xi \right)}^{\frac{1}{{\omega }_2}}.\hfill \end{array}$$

By using the h-convexity of $|{\stackrel{=}{\wedge }}^{″}{|}^{{\omega }_2}$, we can write

$$\begin{array}{cc}\hfill & \le {({𝚤}_2-{𝚤}_1)}^2{\left({\int }_0^1{\left|m\left(\xi \right)\right|}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}\hfill \\ \hfill & \times {\left({\int }_0^1\left(h\left(\xi \right){\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+h(1-\xi ){\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)d\xi \right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \le {({𝚤}_2-{𝚤}_1)}^2{\left({\int }_0^1{\left|m\left(\xi \right)\right|}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}{\left({\int }_0^1\left(M{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+M{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)d\xi \right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \le M{({𝚤}_2-{𝚤}_1)}^2{\left({\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)}^{\frac{1}{{\omega }_2}}{\left({\int }_0^1{\left|m\left(\xi \right)\right|}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}\hfill \\ \hfill & \le M{({𝚤}_2-{𝚤}_1)}^2{\left({\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \times \left({\int }_0^{\frac{1}{2}}{\left|\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{⋋(\frac{\zeta }{k}+1)}-\frac{\xi }{⋋+1}\right|}^{{\omega }_1}d\xi \right.\hfill \\ \hfill & {\left.+{\int }_{\frac{1}{2}}^1{\left|\frac{1-{(1-\xi )}^{\frac{\zeta }{k}+1}-{\xi }^{\frac{\zeta }{k}+1}}{⋋(\frac{\zeta }{k}+1)}-\frac{1-\xi }{⋋+1}\right|}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}\hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{⋋(⋋+1)(\frac{\zeta }{k}+1)}{\left({\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|}^{{\omega }_2}\right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \times \left({\int }_0^{\frac{1}{2}}{\left|⋋+1-(⋋+1)({(1-\xi )}^{\frac{\zeta }{k}+1}+{\xi }^{\frac{\zeta }{k}+1})-⋋(\frac{\zeta }{k}+1)\xi \right|}^{{\omega }_1}d\xi \right.\hfill \\ \hfill & {\left.+{\int }_{\frac{1}{2}}^1{\left|⋋+1-(⋋+1)({(1-\xi )}^{\frac{\zeta }{k}+1}+{\xi }^{\frac{\zeta }{k}+1})-⋋(\frac{\zeta }{k}+1)(1-\xi )\right|}^{{\omega }_1}d\xi \right)}^{\frac{1}{{\omega }_1}}.\hfill \end{array}$$

(27)

However,

$$\begin{array}{cc}\hfill & {\int }_0^{\frac{1}{2}}{\left[⋋+1-(⋋+1)({(1-\xi )}^{\frac{\zeta }{k}+1}+{\xi }^{\frac{\zeta }{k}+1})-⋋(\frac{\zeta }{k}+1)\xi \right]}^{{\omega }_1}d\xi \hfill \\ \hfill & \le \frac{{\left[⋋+1-(⋋+1)2^{-\frac{\zeta }{k}}\right]}^{{\omega }_1+1}-{\left[1+0.5⋋(1-\frac{\zeta }{k})-(1+⋋)2^{-\frac{\zeta }{k}}\right]}^{{\omega }_1+1}}{⋋({\omega }_1+1)(\frac{\zeta }{k}+1)},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\int }_0^{\frac{1}{2}}{\left[-⋋-1+(⋋+1)({(1-\xi )}^{\frac{\zeta }{k}+1}+{\xi }^{\frac{\zeta }{k}+1})+⋋(\frac{\zeta }{k}+1)\xi \right]}^{{\omega }_1}d\xi \hfill \\ \hfill & \le \frac{{\left[⋋(\frac{\zeta }{k}+1)\right]}^{{\omega }_1+1}{2}^{-{\omega }_1-1}}{⋋({\omega }_1+1)(\frac{\zeta }{k}+1)}.\hfill \end{array}$$

Additionally

$$\begin{array}{cc}\hfill & {\int }_{\frac{1}{2}}^1{\left[⋋+1-(⋋+1)({(1-\xi )}^{\frac{\zeta }{k}+1}+{\xi }^{\frac{\zeta }{k}+1})-⋋(\frac{\zeta }{k}+1)(1-\xi )\right]}^{{\omega }_1}d\xi \hfill \\ \hfill & \le \frac{{\left[⋋+1-(⋋+1)2^{-\frac{\zeta }{k}}\right]}^{{\omega }_1+1}-{\left[1+0.5⋋(1-\frac{\zeta }{k})-(⋋+1)2^{-\frac{\zeta }{k}}\right]}^{{\omega }_1+1}}{⋋({\omega }_1+1)(\frac{\zeta }{k}+1)},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\int }_{\frac{1}{2}}^1{\left[-⋋-1+(⋋+1)({(1-\xi )}^{\frac{\zeta }{k}+1}+{\xi }^{\frac{\zeta }{k}+1})+⋋(\frac{\zeta }{k}+1)(1-\xi )\right]}^{{\omega }_1}d\xi \hfill \\ \hfill & \le \frac{{\left[0.5⋋(\frac{\zeta }{k}+1)\right]}^{{\omega }_1+1}}{⋋({\omega }_1+1)(\frac{\zeta }{k}+1)}.\hfill \end{array}$$

Substituting the values of integrals in (27), we obtain the required result. The proof is complete. □

Example 5.

The inequality presented in Theorem 6 can be verified by sketching the graph of (26). For this purpose, by utilizing these expressions (13) and (14) in (23), we obtain

$$\begin{array}{cc}\hfill & \frac{e^{{𝚤}_1}+e^{{𝚤}_2}}{⋋(⋋+1)}+\frac{2}{⋋+1}{e}^{\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)}-\frac{{({𝚤}_2-{𝚤}_1)}^2}{{[⋋(⋋+1)(\frac{\zeta }{k}+1)]}^{1+{\omega }_1^{-1}}}{\left({\left|e^{{𝚤}_1}\right|}^{{\omega }_2}+{\left|e^{{𝚤}_2}\right|}^{{\omega }_2}\right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \times \left(max\left[{\left[⋋+1-(⋋+1)2^{-\frac{\zeta }{k}}\right]}^{{\omega }_1+1}-{\left[1+0.5⋋(1-\frac{\zeta }{k})-(1+⋋)2^{-\frac{\zeta }{k}}\right]}^{{\omega }_1+1},\right.\right.\hfill \\ \hfill & \left.{\left[⋋(\frac{\zeta }{k}+1)\right]}^{{\omega }_1+1}{2}^{-{\omega }_1-1}\right]+max\left[{\left[⋋+1-(⋋+1)2^{-\frac{\zeta }{k}}\right]}^{{\omega }_1+1}\right.\hfill \\ \hfill & {\left.\left.-{\left[1+0.5⋋(1-\frac{\zeta }{k})-(⋋+1)2^{-\frac{\zeta }{k}}\right]}^{{\omega }_1+1},{\left[0.5⋋(\frac{\zeta }{k}+1)\right]}^{{\omega }_1+1}\right]\right)}^{\frac{1}{{\omega }_1}}\hfill \\ \hfill & \le \frac{{\Gamma }_k(\zeta +k)}{k⋋{\Gamma }_k\left(\zeta \right){({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}{\int }_{{𝚤}_1}^{{𝚤}_2}\left[{({𝚤}_2-\xi )}^{\frac{\zeta }{k}-1}+{(\xi -{𝚤}_1)}^{\frac{\zeta }{k}-1}\right]e^{\xi }d\xi \hfill \\ \hfill & \le \frac{e^{{𝚤}_1}+e^{{𝚤}_2}}{⋋(⋋+1)}+\frac{2}{⋋+1}{e}^{\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)}+\frac{{({𝚤}_2-{𝚤}_1)}^2}{{[⋋(⋋+1)(\frac{\zeta }{k}+1)]}^{1+{\omega }_1^{-1}}}{\left({\left|e^{{𝚤}_1}\right|}^{{\omega }_2}+{\left|e^{{𝚤}_2}\right|}^{{\omega }_2}\right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \times \left(max\left[{\left[⋋+1-(⋋+1)2^{-\frac{\zeta }{k}}\right]}^{{\omega }_1+1}-{\left[1+0.5⋋(1-\frac{\zeta }{k})-(1+⋋)2^{-\frac{\zeta }{k}}\right]}^{{\omega }_1+1},\right.\right.\hfill \\ \hfill & \left.{\left[⋋(\frac{\zeta }{k}+1)\right]}^{{\omega }_1+1}{2}^{-{\omega }_1-1}\right]+max\left[{\left[⋋+1-(⋋+1)2^{-\frac{\zeta }{k}}\right]}^{{\omega }_1+1}\right.\hfill \\ \hfill & {\left.\left.-{\left[1+0.5⋋(1-\frac{\zeta }{k})-(⋋+1)2^{-\frac{\zeta }{k}}\right]}^{{\omega }_1+1},{\left[0.5⋋(\frac{\zeta }{k}+1)\right]}^{{\omega }_1+1}\right]\right)}^{\frac{1}{{\omega }_1}}.\hfill \end{array}$$

(28)

Corresponding to the choice of the parameters $M=1,{𝚤}_1=0$, ${𝚤}_2=1$, $⋋=1$, ${\omega }_1=2$, ${\omega }_2=2$, $k=1$, with $1\le \zeta \le 4$, the graph of the double inequality (28) is as below.

$$\begin{array}{cc}\hfill & p_0\left(\zeta \right)=3.5078-\frac{2.0240}{{(\zeta +1)}^{\frac{3}{2}}}.\hfill \\ \hfill & p_1\left(\zeta \right)=\zeta {\int }_0^1\left[{(1-\xi )}^{\zeta -1}+{\xi }^{\zeta -1}\right]e^{\xi }d\xi .\hfill \\ \hfill & p_2\left(\zeta \right)=3.5078+\frac{2.0240}{{(\zeta +1)}^{\frac{3}{2}}}.\hfill \end{array}$$

The numerical values in

Table 5. The comparison results in Example 5 between the double inequality are presented in the following table.

Functions	1	1.5	2	2.5	3	3.5	4
$p_0\left(\zeta \right)$	$2.79221$	$2.99576$	$3.11828$	$3.19869$	$3.2548$	$3.29577$	$3.32677$
$p_1\left(\zeta \right)$	$3.43656$	$3.42856$	$3.43656$	$3.44989$	$3.46454$	$3.47895$	$3.49251$
$p_2\left(\zeta \right)$	$4.22339$	$4.01984$	$3.89732$	$3.81691$	$3.7608$	$3.71983$	$3.68883$

corresponds to Figure 5.

Remark 11.

By substituting $h\left(\xi \right)=\xi$ and $k=1$ in Theorem 6, we obtain ([46] Theorem 5.2) with the choice $s=1.$

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{⋋(⋋+1)}+\frac{2}{⋋+1}\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)-\frac{\Gamma (\zeta +1)}{⋋{({𝚤}_2-{𝚤}_1)}^{\zeta }}\left[{\mathcal{F}}_{{𝚤}_1^{+}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\right|\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{{[⋋(⋋+1)(\zeta +1)]}^{1+{\omega }_1^{-1}}}{\left(\frac{{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}}{2}\right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \times \left(max\left({[⋋+1-(⋋+1)2^{-\zeta }]}^{{\omega }_1+1}+{[1+0.5⋋(1-\zeta )-(1+⋋)2^{-\zeta }]}^{{\omega }_1+1},\right.\right.\hfill \\ \hfill & \left.{[⋋(\zeta +1)]}^{{\omega }_1+1}{2}^{-{\omega }_1-1}\right)\hfill \\ \hfill & +max\left({[⋋+1-(⋋+1)2^{-\zeta }]}^{{\omega }_1+1}-{[1+0.5⋋(1-\zeta )+(1+⋋)2^{-\zeta }]}^{{\omega }_1+1},\right.\hfill \\ \hfill & {\left.\left.{[0.5⋋(\zeta +1)]}^{{\omega }_1+1}\right)\right)}^{\frac{1}{{\omega }_1}},\hfill \end{array}$$

where $\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}=1.$

Remark 12.

By substituting $h\left(\xi \right)={\xi }^s$ and $k=1$ in Theorem 6, we obtain ([46] Theorem 5.2).

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{⋋(⋋+1)}+\frac{2}{⋋+1}\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)-\frac{\Gamma (\zeta +1)}{⋋{({𝚤}_2-{𝚤}_1)}^{\zeta }}\left[{\mathcal{F}}_{{𝚤}_1^{+}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_2\right)+{\mathcal{F}}_{{𝚤}_2^{-}}^{\zeta }\stackrel{=}{\wedge }\left({𝚤}_1\right)\right]\right|\hfill \\ \hfill & \le \frac{{({𝚤}_2-{𝚤}_1)}^2}{{[⋋(⋋+1)(\zeta +1)]}^{1+{\omega }_1^{-1}}}{\left(\frac{{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}+{\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|}^{{\omega }_2}}{s+1}\right)}^{\frac{1}{{\omega }_2}}\hfill \\ \hfill & \times \left[max\left({[⋋+1-(⋋+1)2^{-\zeta }]}^{{\omega }_1+1}+{[1+0.5⋋(1-\zeta )-(1+⋋)2^{-\zeta }]}^{{\omega }_1+1},\right.\right.\hfill \\ \hfill & \left.{[⋋(\zeta +1)]}^{{\omega }_1+1}{2}^{-{\omega }_1-1}\right)\hfill \\ \hfill & +max\left({[⋋+1-(⋋+1)2^{-\zeta }]}^{{\omega }_1+1}-{[1+0.5⋋(1-\zeta )+(1+⋋)2^{-\zeta }]}^{{\omega }_1+1},\right.\hfill \\ \hfill & {\left.\left.{[0.5⋋(\zeta +1)]}^{{\omega }_1+1}\right)\right]}^{\frac{1}{{\omega }_1}},\hfill \end{array}$$

where $\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}=1.$

3. Some Applications to the Main Results in Terms of Means

In mathematics, the means we employ hold profound significance in various domains such as problem-solving, statistical analysis, optimization problems and mathematical proofs. This section contains applications of thhe main results in terms of means. The representation of the means are given as follows:

(i)

The arithmetic mean:

$A=A({𝚤}_1,{𝚤}_2)=\frac{{𝚤}_1+{𝚤}_2}{2},{𝚤}_1,{𝚤}_2\in {\mathbb{R}}^{+}$

(ii)

The logarithmic mean:

$L({𝚤}_1,{𝚤}_2)=\frac{{𝚤}_2-{𝚤}_1}{ln{𝚤}_2-ln{𝚤}_1},{𝚤}_1\ne {𝚤}_2{𝚤}_1,{𝚤}_2\in {\mathbb{R}}^{+}$

Proposition 1.

Let ${𝚤}_1$,${𝚤}_2\in {\mathbb{R}}^{+}$, ${𝚤}_1<{𝚤}_2$; then, we have the following inequalities.

$$\begin{array}{c}\hfill \left|A(e^{{𝚤}_1},e^{{𝚤}_2})-L(e^{{𝚤}_1},e^{{𝚤}_2})\right|\le \frac{{({𝚤}_2-{𝚤}_1)}^2(e^{{𝚤}_1}+e^{{𝚤}_2})}{12}.\end{array}$$

(29)

Proof. 

Using Theorem (1) and making some simplification, we can write this as

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{2}-\frac{k}{2{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}{\int }_{{𝚤}_1}^{{𝚤}_2}[{(y-\xi )}^{\frac{\zeta }{k}-1}+{(\xi -y)}^{\frac{\zeta }{k}-1}]\stackrel{=}{\wedge }\left(\xi \right)d\xi \right|\hfill \\ \hfill & \le M\frac{k{({𝚤}_2-{𝚤}_1)}^2}{2(\zeta +k)}\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)\left(\frac{\zeta }{\zeta +2k}\right).\hfill \end{array}$$

(30)

By substituting $\stackrel{=}{\wedge }\left(\xi \right)=e^{\xi }$, $k=1$, $\zeta =1$ and $M=1$ in (30), we can write this as

$$\begin{array}{cc}\hfill & \left|\frac{e^{{𝚤}_1}+e^{{𝚤}_2}}{2}-\frac{1}{({𝚤}_2-{𝚤}_1)}{\int }_{{𝚤}_1}^{{𝚤}_2}{e}^{\xi }d\xi \right|\le \frac{{({𝚤}_2-{𝚤}_1)}^2(e^{{𝚤}_1}+e^{{𝚤}_2})}{12}\hfill \\ \hfill & \left|\frac{e^{{𝚤}_1}+e^{{𝚤}_2}}{2}-\frac{(e^{{𝚤}_2}-e^{{𝚤}_1})}{({𝚤}_2-{𝚤}_1)}\right|\le \frac{{({𝚤}_2-{𝚤}_1)}^2(e^{{𝚤}_1}+e^{{𝚤}_2})}{12}.\hfill \end{array}$$

This proved relation (29). □

Proposition 2.

Let ${𝚤}_1$,${𝚤}_2\in {\mathbb{R}}^{+}$, ${𝚤}_1<{𝚤}_2$; then, we have the following inequalities.

$$\begin{array}{c}\hfill \left|A(e^{{𝚤}_1},e^{{𝚤}_2})+e^{A(e^{{𝚤}_1},e^{{𝚤}_2})}-2L(e^{{𝚤}_1},e^{{𝚤}_2})\right|\le \frac{{({𝚤}_2-{𝚤}_1)}^2(e^{{𝚤}_1}+e^{{𝚤}_2})}{24}.\end{array}$$

(31)

Proof. 

Using Theorem (5) and making some simplification, we can write this as

$$\begin{array}{cc}\hfill & \left|\frac{\stackrel{=}{\wedge }\left({𝚤}_1\right)+\stackrel{=}{\wedge }\left({𝚤}_2\right)}{⋋(⋋+1)}+\frac{2}{⋋+1}\stackrel{=}{\wedge }\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)-\frac{k}{⋋{({𝚤}_2-{𝚤}_1)}^{\frac{\zeta }{k}}}{\int }_{{𝚤}_1}^{{𝚤}_2}[{(y-\xi )}^{\frac{\zeta }{k}-1}+{(\xi -y)}^{\frac{\zeta }{k}-1}]\stackrel{=}{\wedge }\left(\xi \right)d\xi \right|\hfill \\ \hfill & \le M\frac{{({𝚤}_2-{𝚤}_1)}^2}{⋋(⋋+1)(\frac{\zeta }{k}+1)}\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)\hfill \\ \hfill & \times max\left[\frac{⋋+1}{2}-\frac{⋋(\frac{\zeta }{k}+1)}{8}-\frac{⋋+1}{\frac{\zeta }{k}+2},\frac{-⋋-1}{2}+\frac{⋋(\frac{\zeta }{k}+1)}{8}+\frac{⋋+1}{\frac{\zeta }{k}+2}\right]\hfill \\ \hfill & +M\frac{{({𝚤}_2-{𝚤}_1)}^2}{⋋(⋋+1)(\frac{\zeta }{k}+1)}\left(\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_1\right)\right|+\left|{\stackrel{=}{\wedge }}^{″}\left({𝚤}_2\right)\right|\right)\hfill \\ \hfill & \times max\left[\frac{⋋+1}{2}-\frac{⋋(\frac{\zeta }{k}+1)}{8}-\frac{⋋+1}{\frac{\zeta }{k}+2},\frac{-⋋-1}{2}+\frac{⋋(\frac{\zeta }{k}+1)}{8}+\frac{⋋+1}{\frac{\zeta }{k}+2}\right].\hfill \end{array}$$

(32)

By substituting $\stackrel{=}{\wedge }\left(\xi \right)=e^{\xi }$, $k=1$, $\zeta =1$, $⋋=1$ and $M=1$ in (32), we can write

$$\begin{array}{cc}\hfill & \left|\frac{e^{{𝚤}_1}+e^{{𝚤}_2}}{2}+e^{\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)}-\frac{2}{({𝚤}_2-{𝚤}_1)}{\int }_{{𝚤}_1}^{{𝚤}_2}{e}^{\xi }d\xi \right|\le \frac{{({𝚤}_2-{𝚤}_1)}^2(e^{{𝚤}_1}+e^{{𝚤}_2})}{24}\hfill \\ \hfill & \left|\frac{e^{{𝚤}_1}+e^{{𝚤}_2}}{2}+e^{\left(\frac{{𝚤}_1+{𝚤}_2}{2}\right)}-\frac{2(e^{{𝚤}_2}-e^{{𝚤}_1})}{({𝚤}_2-{𝚤}_1)}\right|\le \frac{{({𝚤}_2-{𝚤}_1)}^2(e^{{𝚤}_1}+e^{{𝚤}_2})}{24}.\hfill \end{array}$$

This completes the proof of (31). □

4. Concluding Remarks

Convexity, a concept that originated from Archimedes around 250 B.C., is a simple and intuitive notion with far-reaching implications in various aspects of our daily lives, including industry, business, medicine and art. Its application is particularly prominent in the field of inequalities, which holds significant importance in optimization theory. In our recent research, we focused on exploring the Hermite–Hadamard integral inequality by employing h-convex functions and a Riemann-type fractional integral. By leveraging Hölder’s inequality, we introduced novel findings that have broad implications for inequality theory. These results were derived based on a newly established identity, allowing us to extend previously published findings and broaden the scope of our investigation. To establish the validity of our obtained results, we represented the double inequalities using graphical representations and tables of values. This comprehensive approach provides concrete evidence supporting our conclusions and further solidifies the significance of our research. Our research serves as a catalyst for future investigations, encouraging researchers to explore more comprehensive outcomes by incorporating generalized fractional operators and expanding the scope of convexities. By embracing these broader perspectives, we anticipate the discovery of more general results that can advance the theory of inequalities and enrich the field of fractional calculus.