A novel class of integral inequalities with graphical approach and diverse applications

ABSTRACT

In this paper, we investigate a novel class of Hermite-Hadamard inequalities applicable to functions with h-convex absolute derivatives. Graphical representations are provided to bolster the validity of our key findings. Some limiting results of our main findings are discussed as corollaries. Furthermore, we establish error estimates in terms of trapezoid formulae for differences between generalized means.

KEYWORDS:

• h-convex function Hermite-Hadamard type inequalities Hölder’s inequality
• MSC 2020:26D10
• 26D15
• 26A33
• 26A51

1. Introduction

Fractional calculus is a mathematical framework that generalizes the traditional operations of differentiation and integration to non-integer orders. This mathematical concept finds application in various fields, including science and engineering (Sun et al. 2018), biomedical engineering (Magin 2004, 2012) and electrochemistry (Oldham 2010). Fractional calculus plays a significant role in inequalities and the analysis of convex functions. Here’s how fractional calculus is related to these concepts: Fractional differential inequalities are inequalities that involve fractional derivatives of functions. These inequalities are essential in analysing the behaviour of solutions to fractional differential equations.

Convex functions are mathematical functions with specific properties, such as non decreasing slopes and being bowl-shaped (Luisa 1990). Fractional calculus can be used to analyse and establish properties of convex functions. In particular, fractional derivatives can be employed to determine the convexity and concavity of functions and provide conditions under which functions are convex.

Inequalities play significant role in establishing these conditions. Inequalities in convex analysis, which is a branch of mathematics that deals with convex sets and functions. Convex analysis has numerous applications across various fields, including mathematics, optimization, economics, engineering and computer science, as discussed in (Niculescu et al. 2006; Nguyen and Tran 2021). Convex analysis and convexity are intimately related concepts in mathematics. Convexity is at the core of convex analysis, providing the foundation for this field. Convex analysis studies and characterizes convexity and associated properties by using several kinds of inequalities. Additionally, Jensen’s inequality plays a pivotal role in probability theory and statistics by demonstrating how the expected value of a convex function compares to the convex function (Dwivedi and Sharma 2022; Fahad et al. 2023). Hölder’s inequality, Young’s inequality and Minkowski’s inequality are also integral to convex analysis, providing tools for handling norms, inner products and integral inequalities (McShane 1937; Gardner 2002; Aldaz 2008; Maligranda 2018; Butt et al. 2022, 2023). The Hermite-Hadamard inequality is a fundamental result in mathematical analysis closely related to convex functions. The mathematician showcased modifications, extensions and refinements of classical inequalities, including the Hermite-Hadamard inequalities (Mitrinovic et al. 1993; Dragomir 2011; Kalsoom et al. 2022; Khan et al. 2022). It establishes an important connection between convexity and integral inequalities and is named after mathematicians Charles Hermite and Jacques Hadamard, as discussed in (Wu et al. 2021; Kashuri et al. 2023). Raissouli et al. explored an algorithm, coupled and recursively refining the Hermite-Hadamard inequality on a simplex to represent the mean value of the integral (Raissouli and Dragomir 2015). Generalized inequalities of the Hermite-Hadamard type for s-convex, $\mathit{s}$-concave and $\mathit{r}$-convex function are derived using classical and Riemann-Liouville fractional integrals (Hudzik and Maligranda 1994; Özdemir et al. 2013, 2016; Feckan et al. 2013; Khan et al. 2017). The brief disussion of several convex functions is explored in (Özdemir et al. 2010, 2011; Zhang et al. 2010; Bai et al. 2012; Wang et al. 2012, 2022; Deng and Wang 2013). The left Riemann-Liouville fractional Hermite-Hadamard inequalities using Green’s function and Jensen’s inequality (Adil Khan et al. 2018) and Schur convexity with Hadamard’s inequality (Chu et al. 2010). Let’s recall the fundamentals used to obtain main results.

In (Varošanec 2007), the $\mathit{h}$-convex function defined as follow:

Definition 1.

A non negative function $\mathit{\gamma }:\mathit{J}\to \mathit{R}$ is called $\mathit{h}$-convex. If ${\zeta }_1,{\zeta }_2\in \mathit{J}$, $\mathit{h}>0$ and $\mathrm{\ell }\in (0,1)$, we have

$\mathit{\gamma }(\mathrm{\ell }{\zeta }_1+(1-\mathrm{\ell }){\zeta }_2)\le \mathit{h}(\mathrm{\ell })\mathit{\gamma }({\zeta }_1)+\mathit{h}(1-\mathrm{\ell })\mathit{\gamma }({\zeta }_2).$

In (Hermite 1883; Hadamard 1893) C. Hermite and J. Hadamard state the following fundamental result which is known as Hermite-Hadamard inequality.

Theorem 1.

Let $\mathit{\gamma }:\mathit{I}\to \mathit{R}$ is a convex function and $[{\zeta }_1,{\zeta }_2]\in \mathit{I}$. Then, the following inequalities hold:

$\mathit{\gamma }\left(\frac{{\zeta }_1+{\zeta }_2}{2}\right)\le \frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathit{\lambda })\mathit{d\lambda }\le \frac{\gamma ({\zeta }_1)+\mathit{\gamma }({\zeta }_2)}{2}.$

The following sections of the paper are organized as follows: In Section 2, we explore the concept of $\mathit{h}$-convexity and examine its fundamental characteristics. In Section 3, we enhance the well-known Hermite-Hadamard inequalities by incorporating the idea of $\mathit{h}$-convexity. We then validate these improved inequalities through graphical representations in both two and three-dimensional spaces. In Section 4, our focus shifts to the exploration of error estimation in generalized means. This exploration effectively demonstrates how the results established in the preceding sections can be practically applied and adapted for various purposes. In the last part, we provide a brief summary of the key discoveries and emphasize the broader significance of our research.

2. Some algebraic properties to $\mathit{h}$-convex function

Algebraic properties hold immense significance in mathematics and various fields beyond. These properties find practical applications in science, engineering, computer science and economics, providing a universal language for problem-solving and modelling real-world phenomena. In particular, the boundedness of convex functions that motivates many desirable properties e.g. optimization, analysis and interpretation in various mathematical and practical contexts. In this section, we will discuss some algebraic properties associated with $\mathit{h}$-convex functions, similar to the algebraic properties of exponential trigonometric convex functions discussed in (Kadakal et al. 2021).

Theorem 2.

Consider an arbitrary family of $\mathit{h}$-convex functions ${\gamma }_{\omega }:\mathit{I}\to \mathbb{R}$ and let $\mathit{\gamma }(\mathit{w})={sup}_{\omega }{\gamma }_{\omega }(\mathit{w})$. If the set $\mathit{J}=[\mathit{w}\in \mathit{I}:({\gamma }_w)<\mathrm{\infty }]$ is not empty, then $\mathit{J}$ forms an interval and the function ℶ is $\mathit{h}$-convex on this interval $\mathit{J}$.

Proof.

Consider $\mathrm{\ell }\in [0,1]$ and let ${\zeta }_1,{\zeta }_2\in \mathit{J}$ be arbitrary. Then

$\mathit{\gamma }(\mathrm{\ell }{\zeta }_1+(1-\mathrm{\ell }){\zeta }_2)=\underset{\omega }{sup}{\gamma }_{\omega }(\mathrm{\ell }{\zeta }_1+(1-\mathrm{\ell }){\zeta }_2)$

$\le \underset{\omega }{sup}\left[\mathrm{\hslash }(\mathit{l}){\gamma }_{\omega }({\zeta }_1)+\mathrm{\hslash }(1-\mathit{l}){\gamma }_{\omega }({\zeta }_2)\right]$

$\le \mathrm{\hslash }(\mathit{l})\underset{\omega }{sup}{\gamma }_{\omega }({\zeta }_1)+\mathrm{\hslash }(1-\mathit{l})\underset{\omega }{sup}{\gamma }_{\omega }({\zeta }_2)$

$=\mathrm{\hslash }(\mathit{l})\mathit{\gamma }({\zeta }_1)+\mathrm{\hslash }(1-\mathit{l})\mathit{\gamma }({\zeta }_2)<\mathrm{\infty }.$

Since the interval $\mathit{J}$ between any two of its points contains every point and ℶ is an $\mathit{h}$-convex function, this completes the proof.□

Theorem 3.

Let ℶ$:[{\zeta }_1,{\zeta }_2]\to \mathit{R}$ be $\mathit{h}$-convex function, then the function ℶ on the interval $[{\zeta }_1,{\zeta }_2]$ is bounded.

Proof.

Let $\mathit{K}=max\{\mathit{\gamma }({\zeta }_1),\mathit{\gamma }({\zeta }_2)\}$ and $\mathit{\nu }\in [{\zeta }_1,{\zeta }_2]$ be an arbitrary point. Then there exist $\mathrm{\ell }\in [0,1]$ such that $\mathit{\nu }=\mathrm{\ell }{\zeta }_1+(1-\mathrm{\ell }){\zeta }_2$. Thus, since $\mathit{h}(\mathrm{\ell })\le 1$ and $\mathit{h}(1-\mathrm{\ell })\le 1$, we have

$\mathit{\gamma }(\mathit{\nu })=\mathit{\gamma }(\mathrm{\ell }{\zeta }_1+(1-\mathrm{\ell }){\zeta }_2)\le \mathrm{\hslash }(\mathit{l})\mathit{\gamma }({\zeta }_1)+\mathrm{\hslash }(1-\mathit{l})({\zeta }_2)\le 2\mathit{K}=\mathit{M}.$

Since

$\mathit{\gamma }(\mathit{\nu })\le \mathit{h}(\mathrm{\ell })\mathit{\gamma }({\zeta }_1)+\mathit{h}(1-\mathrm{\ell })\mathit{\gamma }({\zeta }_2)\le 2\mathit{K}=\mathit{M}.$

Furthermore, there exists a $\mathit{\epsilon }\in [0,\frac{{\zeta }_2-{\zeta }_1}{2}]$, for any $\mathit{\nu }\in [{\zeta }_1,{\zeta }_2]$ such that $\mathit{\nu }=\frac{{\zeta }_1+{\zeta }_2}{2}+\mathit{\epsilon }$ and $\mathit{\nu }=\frac{{\zeta }_1+{\zeta }_2}{2}-\mathit{\epsilon }$. So without loss of generality we can assume, $\mathit{\nu }=\frac{{\zeta }_1+{\zeta }_2}{2}+\mathit{\epsilon }$. Thus, we obtain the following:

$\mathit{\gamma }\left(\frac{{\zeta }_1+{\zeta }_2}{2}\right)=\mathit{\gamma }\left(\frac{1}{2}\left[\frac{{\zeta }_1+{\zeta }_2}{2}+\mathit{\epsilon }\right]+\frac{1}{2}\left[\frac{{\zeta }_1+{\zeta }_2}{2}-\mathit{\epsilon }\right]\right)$

$\le \frac{1}{2}\left(\mathit{\gamma }(\mathit{\nu })+\mathit{\gamma }\left(\frac{{\zeta }_1+{\zeta }_2}{2}-\mathit{\epsilon }\right)\right).$

By considering $\mathit{M}$ to be an upper bound, we get

$\mathit{\gamma }(\mathit{\nu })\ge 2^{\gamma }\left(\frac{{\zeta }_1+{\zeta }_2}{2}\right)-\mathit{\gamma }\left(\frac{{\zeta }_1+{\zeta }_2}{2}-\mathit{\epsilon }\right)\ge 2\mathit{\gamma }\left(\frac{{\zeta }_1+{\zeta }_2}{2}\right)-\mathit{M}=\mathit{m}.$

Therefore, the proof is done.□

Theorem 4.

If the sequence ℶ${ }_n:\mathit{I}\to \mathbb{R}$ consists of convex functions that converge to a finite limit function ℶ on the interval $\mathit{I}$, then ℶ is also convex.

Proof.

Let $\mathrm{\ell }\in (0,1)$ and ${\zeta }_1,{\zeta }_2\in \mathit{I}$,

$\mathit{\gamma }(\mathrm{\ell }{\zeta }_1+(1-\mathrm{\ell }){\zeta }_2)=\underset{\mathit{n}\to \mathrm{\infty }}{lim}{\gamma }_n(\mathrm{\ell }{\zeta }_1+(1-\mathrm{\ell }){\zeta }_2)$

$\le \underset{\mathit{n}\to \mathrm{\infty }}{lim}\left(\mathrm{\hslash }(\mathit{l}){\gamma }_n({\zeta }_1)+\mathrm{\hslash }(1-\mathit{l}){\gamma }_n({\zeta }_2)\right)$

$=\mathrm{\hslash }(\mathit{l})\mathit{\gamma }({\zeta }_1)+\mathrm{\hslash }(1-\mathit{l})\mathit{\gamma }({\zeta }_2).$

So, from above it follows that ℶ is convex.

3. Hermite-Hadamard inequality for $\mathit{h}$-convex function

In this section, we establish our primary findings related to the Hermite-Hadamard inequality for differentiable convex functions. To derive our results, we require the following lemma presented in (Barsam et al. 2021).

Lemma 1.

Let ℶ$:I_0\subseteq \mathit{R}\to \mathit{R}$ be a differentiable function on $I^0$, ${\zeta }_1,{\zeta }_2\in I^0$ such that ${\zeta }_1<{\zeta }_2$. If ℶ${ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}\in \mathit{L}[{\zeta }_1,{\zeta }_2]$ for all $\mathit{\lambda }\in I^0$. Then, we have the following equality.

$\frac{({\zeta }_2-\mathit{\lambda })\mathit{\gamma }({\zeta }_2)+(\mathit{\lambda }-{\zeta }_1)\mathit{\gamma }(\mathit{\zeta })+\mathit{\gamma }(\mathit{\lambda })({\zeta }_2-\mathit{\zeta })}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }$

$=\frac{1}{2({\zeta }_2-{\zeta }_1)}(\mathrm{\ell }-{\mathrm{\ell }}^2)\left[{(\lambda -{\zeta }_1)}^3{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }\mathrm{\ell }+(1-\mathrm{\ell }){\zeta }_1)+{(\lambda -{\zeta }_2)}^3{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }\mathrm{\ell }+(1-\mathrm{\ell }){\zeta }_2)\right].$

Theorem 5.

Let ℶ$:I^{\circ }\subseteq \mathbb{R}\to \mathbb{R}$ is a differentiable function on $I^{\circ }$, where ${\zeta }_1,{\zeta }_2\in I^{\circ }$ such that ${\zeta }_1<{\zeta }_2$. If $|\mathit{\gamma }{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}|$ is $\mathit{h}$-convex function on $[{\zeta }_1,{\zeta }_2]$ for all $\mathit{\lambda }\in I^{\circ }$. Then, we have the following inequality

$\left|\frac{({\zeta }_2-\mathit{\lambda })\mathit{\gamma }({\zeta }_2)+(\mathit{\lambda }-{\zeta }_1)\mathit{\gamma }({\zeta }_1)+\mathit{\gamma }(\mathit{\lambda })({\zeta }_2-{\zeta }_1)}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

(1) $\le \frac{m}{12({\zeta }_2-{\zeta }_1)}\left[|(\mathit{\lambda }-{\zeta }_1{)}^3|\left(|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda })|+|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1)|\right)+|(\mathit{\lambda }-{\zeta }_2{)}^3|\left(|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda })|+|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2)|\right)\right],$(1)

where $\mathit{h}$ is bounded by $\mathit{m}$.

Proof.

By using Lemma 1, we have

$\left|\frac{({\zeta }_2-\mathit{\lambda })\mathit{\gamma }({\zeta }_2)+(\mathit{\lambda }-{\zeta }_1)\mathit{\gamma }({\zeta }_1)+\mathit{\gamma }(\mathit{\lambda })({\zeta }_2-{\zeta }_1)}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_0^1\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2)\left|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_1)\right|\mathit{d}\mathrm{\ell }$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2)\left|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_2)\right|\mathit{d}\mathrm{\ell }.$

By applying $\mathit{h}$-convexity of $|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}|$, we get

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2)\left(\mathit{h}(\mathrm{\ell })|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda })|+\mathit{h}(1-\mathrm{\ell })|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1)|\right)\mathit{d}\mathrm{\ell }$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2)\left(\mathit{h}(\mathrm{\ell })|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda })|+\mathit{h}(1-\mathrm{\ell })|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2)|\right)\mathit{d}\mathrm{\ell }$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2)\left(\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda })|+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1)|\right)\mathit{d}\mathrm{\ell }$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2)\left(\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda })|+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1)|\right)\mathit{d}\mathrm{\ell }$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|\left(\frac{\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda })|+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1)|}{6}\right)$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|\left(\frac{\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda })|+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2)|}{6}\right)$

$\le \frac{m}{12({\zeta }_2-{\zeta }_1)}\left[|(\mathit{\lambda }-{\zeta }_1{)}^3|\left(|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda })|+|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1)|\right)+|(\mathit{\lambda }-{\zeta }_2{)}^3|\left(|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda })|+|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2)|\right)\right].$

Hence, the proof is done.

Example 1.

To validate the inequality presented in Theorem 5, one may inspect the graph of inequality (1). For this analysis, we substitute ℶ$(\mathrm{\ell })=e^{\mathrm{\ell }}$ and obtain the following result.

$-\left[\frac{m}{12({\zeta }_2-{\zeta }_1)}\left(|(\mathit{\lambda }-{\zeta }_1{)}^3|\left(e^{\lambda }+e^{{\zeta }_1}\right)+|(\mathit{\lambda }-{\zeta }_2{)}^3|\left(e^{\lambda }+e^{{\zeta }_2}\right)\right)\right]$

$\le \frac{({\zeta }_2-\mathit{\lambda })e^{{\zeta }_2}+(\mathit{\lambda }-{\zeta }_1)e^{{\zeta }_1}+({\zeta }_2-{\zeta }_1)e^{\lambda }}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}{e}^{\mathrm{\ell }}\mathit{d}\mathrm{\ell }$

(2) $\le \frac{m}{12({\zeta }_2-{\zeta }_1)}\left(|(\mathit{\lambda }-{\zeta }_1{)}^3|\left(e^{\lambda }+e^{{\zeta }_1}\right)+|(\mathit{\lambda }-{\zeta }_2{)}^3|\left(e^{\lambda }+e^{{\zeta }_2}\right)\right).$(2)

Case (i) By setting the parameters $\mathit{m}=1$ and $\mathit{\lambda }=1$ with $\mathit{L}({\zeta }_1,{\zeta }_2)$, $\mathit{M}({\zeta }_1,{\zeta }_2)$ and $\mathit{R}({\zeta }_1,{\zeta }_2)$ denoting the left, middle and right components and considering the ranges of ${\zeta }_1\in [5,6]$ and ${\zeta }_2\in [7,8]$, the graph of the double inequality (2) can be presented as follows.

Case (ii) By setting the parameters as $\mathit{m}=1$, ${\zeta }_1=1$ and ${\zeta }_2=2$ and specifying the functions $\mathit{L}(\mathit{\lambda })$, $\mathit{M}(\mathit{\lambda })$ and $\mathit{R}(\mathit{\lambda })$ as the left, middle and right components and while examining the range of $\mathit{\lambda }\in [4,8]$, the graph of the double inequality (2) can be presented as follows.

The Figures 1,2 visually confirm the results obtained in 3D and 2D graphs of Equation (2)(2) $\le \frac{m}{12({\zeta }_2-{\zeta }_1)}\left(|(\mathit{\lambda }-{\zeta }_1{)}^3|\left(e^{\lambda }+e^{{\zeta }_1}\right)+|(\mathit{\lambda }-{\zeta }_2{)}^3|\left(e^{\lambda }+e^{{\zeta }_2}\right)\right).$(2) .

Figure 1. The graph that show the inequality (2) corresponds to the above mentioned parameters.

Figure 2. The graph that show the inequality (2) corresponds to the above mentioned parameters.

Corollary 1.

With the assumptions that ℶ$:I^{\circ }\subseteq \mathbb{R}\to \mathbb{R}$ is a differentiable function on $I^{\circ }$ and ${\zeta }_1,{\zeta }_2\in I^{\circ }$ such that ${\zeta }_1<{\zeta }_2$. If $|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}|$ is $\mathit{h}$-convex function on $[{\zeta }_1,{\zeta }_2]$. If we choose $\mathit{\lambda }={\zeta }_1$, we obtain the following limiting result.

$\left|\frac{\gamma ({\zeta }_1)+\mathit{\gamma }({\zeta }_2)}{2}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|\le \frac{m{({\zeta }_1-{\zeta }_2)}^2\left(|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1)|+|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2)|\right)}{12}.$

Corollary 2.

If we choose $\mathit{\lambda }=\frac{{\zeta }_1+{\zeta }_2}{2}$ in Theorem 5, then we have the following limiting relation

$\left|\frac{\gamma ({\zeta }_1)+\mathit{\gamma }({\zeta }_2)+2\mathit{\gamma }\left(\frac{{\zeta }_1+{\zeta }_2}{2}\right)}{2}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

$\le \frac{m{({\zeta }_1-{\zeta }_2)}^2}{96}\left(|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1)|+2{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}\left(\frac{{\zeta }_1+{\zeta }_2}{2}\right)+|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2)|\right).$

Theorem 6.

Let ℶ$:I^{\circ }\subseteq \mathbb{R}\to \mathbb{R}$ is a differentiable function on $I^{\circ }$, where ${\zeta }_1,{\zeta }_2\in I^{\circ }$ such that ${\zeta }_1<{\zeta }_2$. If $|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}{|}^{{\beta }_2}$ is $\mathit{h}$-convex function on $[{\zeta }_1,{\zeta }_2]$ and ${\beta }_2>1$. Then, we have the following inequality

$\left|\frac{({\zeta }_2-\mathit{\lambda })\mathit{\gamma }({\zeta }_2)+(\mathit{\lambda }-{\zeta }_1)\mathit{\gamma }({\zeta }_1)+\mathit{\gamma }(\mathit{\lambda })({\zeta }_2-{\zeta }_1)}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{1}{6}\right)}^{1-\frac{1}{{\beta }_2}}{\left(\frac{\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}{({\zeta }_1)}^{{\beta }_2}|}{6}\right)}^{\frac{1}{{\beta }_2}}$

(3) $+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{1}{6}\right)}^{1-\frac{1}{{\beta }_2}}{\left(\frac{\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}{({\zeta }_2)}^{{\beta }_2}|}{6}\right)}^{\frac{1}{{\beta }_2}},$(3)

where $\mathit{h}$ is bounded by $\mathit{m}$.

Proof.

By using Lemma 1, we have

$\left|\frac{({\zeta }_2-\mathit{\lambda })\mathit{\gamma }({\zeta }_2)+(\mathit{\lambda }-{\zeta }_1)\mathit{\gamma }({\zeta }_1)+\mathit{\gamma }(\mathit{\lambda })({\zeta }_2-{\zeta }_1)}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2{)}^{1-\frac{1}{{\beta }_2}}(\mathrm{\ell }-{\mathrm{\ell }}^2{)}^{\frac{1}{{\beta }_2}}\left|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_1)\right|\mathit{d}\mathrm{\ell }$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2{)}^{1-\frac{1}{{\beta }_2}}(\mathrm{\ell }-{\mathrm{\ell }}^2{)}^{\frac{1}{{\beta }_2}}\left|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_2)\right|\mathit{d}\mathrm{\ell }$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2{)}^{1-\frac{1}{{\beta }_2}}{\left({\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2)|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_1{)|}^{{\beta }_2}\mathit{d}\mathrm{\ell }\right)}^{\frac{1}{{\beta }_2}}$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2{)}^{1-\frac{1}{{\beta }_2}}{\left({\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2)|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_2{)|}^{{\beta }_2}\mathit{d}\mathrm{\ell }\right)}^{\frac{1}{{\beta }_2}}.$

By applying $\mathit{h}$-convexity of $|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}|$, we get

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2{)}^{1-\frac{1}{{\beta }_2}}{\left({\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2)(\mathit{h}(\mathrm{\ell })|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda })|+\mathit{h}(1-\mathrm{\ell })|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1{)|)|}^{{\beta }_2}\mathit{d}\mathrm{\ell }\right)}^{\frac{1}{{\beta }_2}}$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2{)}^{1-\frac{1}{{\beta }_2}}{\left({\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2)(\mathit{h}(\mathrm{\ell })|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda })|+\mathit{h}(1-\mathrm{\ell })|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2{)|)}^{{\beta }_2}\mathit{d}\mathrm{\ell }\right)}^{\frac{1}{{\beta }_2}}$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{1}{6}\right)}^{1-\frac{1}{{\beta }_2}}{\left(\frac{\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1{)|}^{{\beta }_2}}{6}\right)}^{\frac{1}{{\beta }_2}}$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{1}{6}\right)}^{1-\frac{1}{{\beta }_2}}{\left(\frac{\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2{)|}^{{\beta }_2}}{6}\right)}^{\frac{1}{{\beta }_2}}.$

Hence, the proof is done.

Example 2.

To validate the inequality presented in Theorem 6, one may inspect the graph of inequality (3). For this analysis, we substitute $\mathit{\gamma }(\mathrm{\ell })=e^{\mathrm{\ell }}$ and obtain the following result.

$-\left[\left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{1}{6}\right)}^{1-\frac{1}{{\beta }_2}}{\left(\frac{\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_1}{|}^{{\beta }_2}}{6}\right)}^{\frac{1}{{\beta }_2}}\right.$

$\left.+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{1}{6}\right)}^{1-\frac{1}{{\beta }_2}}{\left(\frac{\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_2}{|}^{{\beta }_2}}{6}\right)}^{\frac{1}{{\beta }_2}}\right]$

$\le \frac{({\zeta }_2-\mathit{\lambda }){{\zeta }_2}^2+(\mathit{\lambda }-{\zeta }_1){{\zeta }_1}^2+({\zeta }_2-{\zeta }_1){\lambda }^2}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}{\mathrm{\ell }}^2\mathit{d}\mathrm{\ell }$

$\le \left[\left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{1}{6}\right)}^{1-\frac{1}{{\beta }_2}}{\left(\frac{\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_1}{|}^{{\beta }_2}}{6}\right)}^{\frac{1}{{\beta }_2}}\right.$

(4) $\left.+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{1}{6}\right)}^{1-\frac{1}{{\beta }_2}}{\left(\frac{\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_2}{|}^{{\beta }_2}}{6}\right)}^{\frac{1}{{\beta }_2}}\right].$(4)

Case (i) By setting the parameters $\mathit{m}=1$ and $\mathit{\lambda }=1$ with $\mathit{L}({\zeta }_1,{\zeta }_2)$, $\mathit{M}({\zeta }_1,{\zeta }_2)$ and $\mathit{R}({\zeta }_1,{\zeta }_2)$ denoting the left, middle and right components and considering the ranges of ${\zeta }_1\in [1,5]$ and ${\zeta }_2\in [6,10]$, the graph of the double inequality (4) can be presented as follows.

Case (ii) By setting the parameters as $\mathit{m}=1$, ${\zeta }_1=1$ and ${\zeta }_2=2$ and specifying the functions $\mathit{L}(\mathit{\lambda })$, $\mathit{M}(\mathit{\lambda })$ and $\mathit{R}(\mathit{\lambda })$ as the left, middle and right components and while examining the range of $\mathit{\lambda }\in [1,2]$, the graph of the double inequality (4) can be presented as follows.

The Figures 3,4 visually confirm the results obtained in 3D and 2D graphs of Equation (3)(3) $+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{1}{6}\right)}^{1-\frac{1}{{\beta }_2}}{\left(\frac{\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}{({\zeta }_2)}^{{\beta }_2}|}{6}\right)}^{\frac{1}{{\beta }_2}},$(3) .

Figure 3. The graph that show the inequality (4) corresponds to the above mentioned parameters.

Figure 4. The graph that show the inequality (4) corresponds to the above mentioned parameters.

Corollary 3.

With the assumption that $\mathit{\gamma }:I^{\circ }\subseteq \mathbb{R}\to \mathbb{R}$ is a differentiable function on $I^{\circ }$ and ${\zeta }_1,{\zeta }_2\in I^{\circ }$ such that ${\zeta }_1<{\zeta }_2$. If $|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}{|}^{{\beta }_2}$ is $\mathit{h}$-convex function on $[{\zeta }_1,{\zeta }_2]$ and ${\beta }_2>1$. If we choose $\mathit{\lambda }={\zeta }_1$, we get

$\left|\frac{\gamma ({\zeta }_1)+\mathit{\gamma }({\zeta }_2)}{2}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

$\le \frac{{({\zeta }_2-{\zeta }_1)}^2}{2}{\left(\frac{1}{6}\right)}^{1-\frac{1}{{\beta }_2}}{\left(\frac{\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}{({\zeta }_2)}^{{\beta }_2}|}{6}\right)}^{\frac{1}{{\beta }_2}}.$

Corollary 4.

If we put $\mathit{\lambda }=\frac{{\zeta }_1+{\zeta }_2}{2}$ in Theorem 6, then we have

$\left|\frac{\gamma ({\zeta }_1)+\mathit{\gamma }({\zeta }_2)+2\mathit{\gamma }\left(\frac{{\zeta }_1+{\zeta }_2}{2}\right)}{4}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

$\le \frac{{({\zeta }_2-{\zeta }_1)}^2}{16}{\left(\frac{1}{6}\right)}^{1-\frac{1}{{\beta }_2}}{\left(\frac{\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\frac{{\zeta }_1+{\zeta }_2}{2}{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}{({\zeta }_1)}^{{\beta }_2}|}{6}\right)}^{\frac{1}{{\beta }_2}}$

$+\frac{{({\zeta }_2-{\zeta }_1)}^2}{16}{\left(\frac{1}{6}\right)}^{1-\frac{1}{{\beta }_2}}{\left(\frac{\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\frac{{\zeta }_1+{\zeta }_2}{2}{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}{({\zeta }_2)}^{{\beta }_2}|}{6}\right)}^{\frac{1}{{\beta }_2}}.$

Theorem 7.

Let $\mathit{\gamma }:\mathit{I}\subseteq (0,+\mathrm{\infty }]\to \mathbb{R}$ is a differentiable function on $I^{\circ }$ with ${\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}\in \mathit{L}[{\zeta }_1,{\zeta }_2]$, where ${\zeta }_1,{\zeta }_2\in \mathit{I}$ such that ${\zeta }_1<{\zeta }_2$. If $|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}|$ is $\mathit{h}$-convex function on $[{\zeta }_1,{\zeta }_2]$ and ${\beta }_2>1$ with $\frac{1}{{\beta }_1}+\frac{1}{{\beta }_2}=1$. Then, we have the following inequality

$\left|\frac{({\zeta }_2-\mathit{\lambda })\mathit{\gamma }({\zeta }_2)+(\mathit{\lambda }-{\zeta }_1)\mathit{\gamma }({\zeta }_1)+\mathit{\gamma }(\mathit{\lambda })({\zeta }_2-{\zeta }_1)}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{8({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{\mathrm{\Gamma }({\beta }_1+1)}{\mathrm{\Gamma }({\beta }_1+\frac{3}{2})}\right)}^{\frac{1}{{\beta }_1}}{\left(\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1{)|}^{{\beta }_2}\right)}^{\frac{1}{{\beta }_2}}$

(5) $+\left|\frac{{(\lambda -{\zeta }_2)}^3}{8({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{\mathrm{\Gamma }({\beta }_1+1)}{\mathrm{\Gamma }({\beta }_1+\frac{3}{2})}\right)}^{\frac{1}{{\beta }_1}}{\left(\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2{)|}^{{\beta }_2}\right)}^{\frac{1}{{\beta }_2}},$(5)

where $\mathit{h}$ is bounded by $\mathit{m}$.

Proof.

By using Lemma 1, we have

$\left|\frac{({\zeta }_2-\mathit{\lambda })\mathit{\gamma }({\zeta }_2)+(\mathit{\lambda }-{\zeta }_1)\mathit{\gamma }({\zeta }_1)+(\mathit{\lambda })({\zeta }_2-{\zeta }_1)}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2)\left|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_1)\right|\mathit{d}\mathrm{\ell }$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2)\left|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_2)\right|\mathit{d}\mathrm{\ell }$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left({\int }_0^1{\mathrm{\ell }}^{{\beta }_1}{(1-\mathrm{\ell })}^{{\beta }_1}\right)}^{\frac{1}{{\beta }_1}}{\left({\int }_0^1{\left|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_1)\right|}^{{\beta }_2}\mathit{d}\mathrm{\ell }\right)}^{\frac{1}{{\beta }_2}}$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left({\int }_0^1{\mathrm{\ell }}^{{\beta }_1}{(1-\mathrm{\ell })}^{{\beta }_1}\right)}^{\frac{1}{{\beta }_1}}{\left({\int }_0^1{\left|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_2)\right|}^{{\beta }_2}\mathit{d}\mathrm{\ell }\right)}^{\frac{1}{{\beta }_2}},$

on the other hand

${\int }_0^1{\left|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_2)\right|}^{{\beta }_2}\mathit{d}\mathrm{\ell }\le |\mathit{h}(\mathrm{\ell })|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }){|}^{{\beta }_2}+\mathit{h}(1-\mathrm{\ell })|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1){|}^{{\beta }_2}\mathit{d}\mathrm{\ell }.$

Substituting $\mathit{h}(\mathrm{\ell })=\mathit{h}(1-\mathrm{\ell })=\mathit{m}$

${\int }_0^1{\left|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_2)\right|}^{{\beta }_2}\mathit{d}\mathrm{\ell }\le \mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }){|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1){|}^{{\beta }_2}.$

Also

$\mathit{B}({\beta }_1,{\beta }_1)={\int }_0^1{\mathrm{\ell }}^{{\beta }_1-1}(1-\mathrm{\ell }{)}^{{\beta }_1-1}\mathit{d}\mathrm{\ell },{\beta }_1>0,$

and

$\mathit{B}({\beta }_1+1,{\beta }_1+1)=2^{1-2({\beta }_1+1)}\mathit{B}\left(\frac{1}{2},{\beta }_1+1\right)=\frac{2^{1-2({\beta }_1+1)}\mathrm{\Gamma }(\frac{1}{2})\mathrm{\Gamma }({\beta }_1+1)}{\mathrm{\Gamma }({\beta }_1+\frac{3}{2})}.$

So

$\left|\frac{({\zeta }_2-\mathit{\lambda })\mathit{\gamma }({\zeta }_2)+(\mathit{\lambda }-{\zeta }_1)\mathit{\gamma }({\zeta }_1)+\mathit{\gamma }(\mathit{\lambda })({\zeta }_2-{\zeta }_1)}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{\sqrt{\mathit{\pi }}}{2}\right)}^{1-\frac{1}{{\beta }_1}}{\left(\frac{2^{1-2({\beta }_1+1)}\mathrm{\Gamma }(\frac{1}{2})\mathrm{\Gamma }({\beta }_1+1)}{\mathrm{\Gamma }({\beta }_1+\frac{3}{2})}\right)}^{\frac{1}{{\beta }_1}}{\left(\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1{)|}^{{\beta }_2}\right)}^{\frac{1}{{\beta }_2}}$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{\sqrt{\mathit{\pi }}}{2}\right)}^{1-\frac{1}{{\beta }_1}}{\left(\frac{2^{1-2({\beta }_1+1)}\mathrm{\Gamma }(\frac{1}{2})\mathrm{\Gamma }({\beta }_1+1)}{\mathrm{\Gamma }({\beta }_1+\frac{3}{2})}\right)}^{\frac{1}{{\beta }_1}}{\left(\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2{)|}^{{\beta }_2}\right)}^{\frac{1}{{\beta }_2}}$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{8({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{\mathrm{\Gamma }({\beta }_1+1)}{\mathrm{\Gamma }({\beta }_1+\frac{3}{2})}\right)}^{\frac{1}{{\beta }_1}}{\left(\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1{)|}^{{\beta }_2}\right)}^{\frac{1}{{\beta }_2}}$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{8({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{\mathrm{\Gamma }({\beta }_1+1)}{\mathrm{\Gamma }({\beta }_1+\frac{3}{2})}\right)}^{\frac{1}{{\beta }_1}}{\left(\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2{)|}^{{\beta }_2}\right)}^{\frac{1}{{\beta }_2}}.$

This completes the proof of the result.

Example 3.

To validate the inequality presented in Theorem 7, one may inspect the graph of inequality (5). For this analysis, we substitute $\mathit{\gamma }(\mathrm{\ell })=e^{\mathrm{\ell }}$ and obtain the following result.

$-\left[\left|\frac{{(\lambda -{\zeta }_1)}^3}{8({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{\mathrm{\Gamma }({\beta }_1+1)}{\mathrm{\Gamma }({\beta }_1+\frac{3}{2})}\right)}^{\frac{1}{{\beta }_1}}{\left(\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_1}{|}^{{\beta }_2}\right)}^{\frac{1}{{\beta }_2}}\right.$

$\left.+\left|\frac{{(\lambda -{\zeta }_2)}^3}{8({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{\mathrm{\Gamma }({\beta }_1+1)}{\mathrm{\Gamma }({\beta }_1+\frac{3}{2})}\right)}^{\frac{1}{{\beta }_1}}{\left(\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_2}{|}^{{\beta }_2}\right)}^{\frac{1}{{\beta }_2}}\right]$

$\le \frac{({\zeta }_2-\mathit{\lambda })e^{{\zeta }_2}+(\mathit{\lambda }-{\zeta }_1)e^{{\zeta }_1}+({\zeta }_2-{\zeta }_1)e^{\lambda }}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}{e}^{\mathrm{\ell }}\mathit{d}\mathrm{\ell }$

$\le \left[\left|\frac{{(\lambda -{\zeta }_1)}^3}{8({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{\mathrm{\Gamma }({\beta }_1+1)}{\mathrm{\Gamma }({\beta }_1+\frac{3}{2})}\right)}^{\frac{1}{{\beta }_1}}{\left(\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_1}{|}^{{\beta }_2}\right)}^{\frac{1}{{\beta }_2}}\right.$

(6) $\left.+\left|\frac{{(\lambda -{\zeta }_2)}^3}{8({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{\mathrm{\Gamma }({\beta }_1+1)}{\mathrm{\Gamma }({\beta }_1+\frac{3}{2})}\right)}^{\frac{1}{{\beta }_1}}{\left(\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_2}{|}^{{\beta }_2}\right)}^{\frac{1}{{\beta }_2}}\right].$(6)

Case (i) By setting the parameters $\mathit{m}=1$ and $\mathit{\lambda }=1$ with $\mathit{L}({\zeta }_1,{\zeta }_2)$, $\mathit{M}({\zeta }_1,{\zeta }_2)$ and $\mathit{R}({\zeta }_1,{\zeta }_2)$ denoting the left, middle and right components and considering the ranges of ${\zeta }_1\in [11,15]$ and ${\zeta }_2\in [16,20]$, the graph of the double inequality (6) can be presented as follows.

Case (ii) By setting the parameters as $\mathit{m}=1$, ${\zeta }_1=1$ and ${\zeta }_2=2$ and specifying the functions $\mathit{L}(\mathit{\lambda })$, $\mathit{M}(\mathit{\lambda })$ and $\mathit{R}(\mathit{\lambda })$ as the left, middle and right components and while examining the range of $\mathit{\lambda }\in [3,5]$, the graph of the double inequality (6) can be presented as follows.

The Figures 5,6 visually confirm the results obtained in 3D and 2D graphs of Equation (6)(6) $\left.+\left|\frac{{(\lambda -{\zeta }_2)}^3}{8({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{\mathrm{\Gamma }({\beta }_1+1)}{\mathrm{\Gamma }({\beta }_1+\frac{3}{2})}\right)}^{\frac{1}{{\beta }_1}}{\left(\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_2}{|}^{{\beta }_2}\right)}^{\frac{1}{{\beta }_2}}\right].$(6) .

Figure 5. The graph that show the inequality (6) corresponds to the above mentioned parameters.

Figure 6. The graph that show the inequality (6) corresponds to the above mentioned parameters.

Corollary 5.

Suppose that $\mathit{\gamma }:\mathit{I}\subseteq (0,+\mathrm{\infty }]\to \mathbb{R}$ is a differentiable function on $I^{\circ }$ with ${\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}\in \mathit{L}[{\zeta }_1,{\zeta }_2]$, where ${\zeta }_1,{\zeta }_2\in \mathit{I}$ such that ${\zeta }_1<{\zeta }_2$. And with the assumption that $|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}|$ is $\mathit{h}$-convex function on $[{\zeta }_1,{\zeta }_2]$ and ${\beta }_2>1$ with $\frac{1}{{\beta }_1}+\frac{1}{{\beta }_2}=1$. If we choose $\mathit{\lambda }$ as ${\zeta }_1$ or ${\zeta }_2$. Then, we get the following inequality

$\left|\frac{({\zeta }_2-{\zeta }_1)\mathit{\gamma }({\zeta }_2)+({\zeta }_1-{\zeta }_1)\mathit{\gamma }({\zeta }_1)+\mathit{\gamma }({\zeta }_1)({\zeta }_2-{\zeta }_1)}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

$\le \frac{m{(\lambda -{\zeta }_2)}^3}{2}{\left(\frac{\mathrm{\Gamma }({\beta }_1+1)}{\mathrm{\Gamma }({\beta }_1+\frac{3}{2})}\right)}^{\frac{1}{{\beta }_1}}{\left(|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2{)|}^{{\beta }_2}\right)}^{\frac{1}{{\beta }_2}}.$

Corollary 6.

If we choose $\mathit{\lambda }=\frac{{\zeta }_1+{\zeta }_2}{2}$ in Theorem 7, then we have

$\left|\frac{\gamma ({\zeta }_1)+\mathit{\gamma }({\zeta }_2)+2\mathit{\gamma }\left(\frac{{\zeta }_1+{\zeta }_2}{2}\right)}{4}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

$\le \frac{{({\zeta }_2-{\zeta }_1)}^2}{64}{\left(\frac{\mathrm{\Gamma }({\beta }_1+1)}{\mathrm{\Gamma }({\beta }_1+\frac{3}{2})}\right)}^{\frac{1}{{\beta }_1}}{\left(\mathit{m}{\left|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}\left(\frac{{\zeta }_1+{\zeta }_2}{2}\right)\right|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1{)|}^{{\beta }_2}\right)}^{\frac{1}{{\beta }_2}}$

$+\frac{{({\zeta }_2-{\zeta }_1)}^2}{64}{\left(\frac{\mathrm{\Gamma }({\beta }_1+1)}{\mathrm{\Gamma }({\beta }_1+\frac{3}{2})}\right)}^{\frac{1}{{\beta }_1}}{\left(\mathit{m}{\left|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}\left(\frac{{\zeta }_1+{\zeta }_2}{2}\right)\right|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2{)|}^{{\beta }_2}\right)}^{\frac{1}{{\beta }_2}}.$

Theorem 8.

Let $\mathit{\gamma }:\mathit{I}\subseteq (0,+\mathrm{\infty }]\to \mathbb{R}$ is a differentiable function on $I^{\circ }$ with ${\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}\in \mathit{L}[{\zeta }_1,{\zeta }_2]$, where ${\zeta }_1,{\zeta }_2\in \mathit{I}$ such that ${\zeta }_1<{\zeta }_2$. If $|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}|$ is $\mathit{h}$-convex function on $[{\zeta }_1,{\zeta }_2]$ and ${\beta }_2>1$ with $\frac{1}{{\beta }_1}+\frac{1}{{\beta }_2}=1$. Then, we have the following inequality

$\left|\frac{({\zeta }_2-\mathit{\lambda })\mathit{\gamma }({\zeta }_2)+(\mathit{\lambda }-{\zeta }_1)\mathit{\gamma }({\zeta }_1)+\mathit{\gamma }(\mathit{\lambda })({\zeta }_2-{\zeta }_1)}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{1}{{\beta }_1+1}\right)}^{\frac{1}{{\beta }_1}}{\left(\frac{\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1{)|}^{{\beta }_2}}{{\beta }_2+1}\right)}^{\frac{1}{{\beta }_2}}$

(7) $+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{1}{{\beta }_1+1}\right)}^{\frac{1}{{\beta }_1}}{\left(\frac{\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2{)|}^{{\beta }_2}}{{\beta }_2+1}\right)}^{\frac{1}{{\beta }_2}},$(7)

where $\mathit{h}$ is bounded by $\mathit{m}$.

Proof.

By using Lemma 1, we have

$\left|\frac{({\zeta }_2-\mathit{\lambda })\mathit{\gamma }({\zeta }_2)+(\mathit{\lambda }-{\zeta }_1)\mathit{\gamma }({\zeta }_1)+\mathit{\gamma }(\mathit{\lambda })({\zeta }_2-{\zeta }_1)}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2)\left|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_1)\right|\mathit{d}\mathrm{\ell }$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2)\left|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_2)\right|\mathit{d}\mathrm{\ell }$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left({\int }_0^1{\mathrm{\ell }}^{{\beta }_1}\right)}^{\frac{1}{{\beta }_1}}{\left({\int }_0^1{(1-\mathrm{\ell })}^{{\beta }_2}{\left|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_1)\right|}^{{\beta }_2}\mathit{d}\mathrm{\ell }\right)}^{\frac{1}{{\beta }_2}}$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left({\int }_0^1{\mathrm{\ell }}^{{\beta }_1}\right)}^{\frac{1}{{\beta }_1}}{\left({\int }_0^1{(1-\mathrm{\ell })}^{{\beta }_2}{\left|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_2)\right|}^{{\beta }_2}\mathit{d}\mathrm{\ell }\right)}^{\frac{1}{{\beta }_2}}.$

By applying $\mathit{h}$-convexity of $|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}|$, we get

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left({\int }_0^1{\mathrm{\ell }}^{{\beta }_1}\right)}^{\frac{1}{{\beta }_1}}{\left({\int }_0^1{(1-\mathrm{\ell })}^{{\beta }_2}(\mathit{h}(\mathrm{\ell })|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{h}(1-\mathrm{\ell })|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1{)|}^{{\beta }_2}\mathit{d}\mathrm{\ell })\right)}^{\frac{1}{{\beta }_2}}$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left({\int }_0^1{\mathrm{\ell }}^{{\beta }_1}\right)}^{\frac{1}{{\beta }_1}}{\left({\int }_0^1{(1-\mathrm{\ell })}^{{\beta }_2}{(\mathit{h}(\mathrm{\ell })|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{h}(1-\mathrm{\ell })|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2{)|}^{{\beta }_2})}^{{\beta }_2}\mathit{d}\mathrm{\ell }\right)}^{\frac{1}{{\beta }_2}}.$

By substituting $\mathit{h}(\mathrm{\ell })=\mathit{h}(1-\mathrm{\ell })=\mathit{m}$, we obtain the required result.□

Example 4.

To validate the inequality presented in Theorem 8, one may inspect the graph of inequality (7). For this analysis, we substitute $\mathit{\gamma }(\mathrm{\ell })=e^{\mathrm{\ell }}$ and obtain the following result.

$-\left[\left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{1}{{\beta }_1+1}\right)}^{\frac{1}{{\beta }_1}}{\left(\frac{\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_1}{|}^{{\beta }_2}}{{\beta }_2+1}\right)}^{\frac{1}{{\beta }_2}}\right.$

$+\left.\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{1}{{\beta }_1+1}\right)}^{\frac{1}{{\beta }_1}}{\left(\frac{\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_2}{|}^{{\beta }_2}}{{\beta }_2+1}\right)}^{\frac{1}{{\beta }_2}}\right]$

$\le \frac{({\zeta }_2-\mathit{\lambda }){{\zeta }_2}^2+(\mathit{\lambda }-{\zeta }_1){{\zeta }_1}^2+({\zeta }_2-{\zeta }_1){\lambda }^2}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}{\mathrm{\ell }}^2\mathit{d}\mathrm{\ell }$

$\le \left[\left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{1}{{\beta }_1+1}\right)}^{\frac{1}{{\beta }_1}}{\left(\frac{\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_1}{|}^{{\beta }_2}}{{\beta }_2+1}\right)}^{\frac{1}{{\beta }_2}}\right.$

(8) $+\left.\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{1}{{\beta }_1+1}\right)}^{\frac{1}{{\beta }_1}}{\left(\frac{\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_2}{|}^{{\beta }_2}}{{\beta }_2+1}\right)}^{\frac{1}{{\beta }_2}}\right].$(8)

Case (i) By setting the parameters $\mathit{m}=1$ and $\mathit{\lambda }=1$ with $\mathit{L}({\zeta }_1,{\zeta }_2)$, $\mathit{M}({\zeta }_1,{\zeta }_2)$ and $\mathit{R}({\zeta }_1,{\zeta }_2)$ denoting the left, middle and right components and considering the ranges of ${\zeta }_1\in [11,15]$ and ${\zeta }_2\in [16,20]$, the graph of the double inequality (8) can be presented as follows.

Case (ii) By setting the parameters as $\mathit{m}=1$, ${\zeta }_1=1$ and ${\zeta }_2=2$ and specifying the functions $\mathit{L}(\mathit{\lambda })$, $\mathit{M}(\mathit{\lambda })$ and $\mathit{R}(\mathit{\lambda })$ as the left, middle and right components and while examining the range of $\mathit{\lambda }\in [0,4]$, the graph of the double inequality (8) can be presented as follows.

The Figures 7,8 visually confirm the results obtained in 3D and 2D graphs of Equation (8)(8) $+\left.\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{1}{{\beta }_1+1}\right)}^{\frac{1}{{\beta }_1}}{\left(\frac{\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_2}{|}^{{\beta }_2}}{{\beta }_2+1}\right)}^{\frac{1}{{\beta }_2}}\right].$(8) .

Figure 7. The graph that show the inequality (8) corresponds to the above mentioned parameters.

Figure 8. The graph that show the inequality (8) corresponds to the above mentioned parameters.

Theorem 9.

Let $\mathit{\gamma }:\mathit{I}\subseteq (0,+\mathrm{\infty }]\to \mathbb{R}$ is a differentiable function on $I^{\circ }$ with ${\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}\in \mathit{L}[{\zeta }_1,{\zeta }_2]$, where ${\zeta }_1,{\zeta }_2\in \mathit{I}$ such that ${\zeta }_1<{\zeta }_2$. If $|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}|$ is $\mathit{h}$-convex function on $[{\zeta }_1,{\zeta }_2]$ and ${\beta }_2>1$ with $\frac{1}{{\beta }_1}+\frac{1}{{\beta }_2}=1$. Then, we have the following inequality

$\left|\frac{({\zeta }_2-\mathit{\lambda })\mathit{\gamma }({\zeta }_2)+(\mathit{\lambda }-{\zeta }_1)\mathit{\gamma }({\zeta }_1)+\mathit{\gamma }(\mathit{\lambda })({\zeta }_2-{\zeta }_1)}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(B(2,{\beta }_2+1)(\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1{)|}^{{\beta }_2})\right)}^{\frac{1}{{\beta }_2}}$

(9) $+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(B(2,{\beta }_2+1)(\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2{)|}^{{\beta }_2})\right)}^{\frac{1}{{\beta }_2}},$(9)

where $\mathit{h}$ is bounded by $\mathit{m}$.

Proof.

By using Lemma 1, we have

$\left|\frac{({\zeta }_2-\mathit{\lambda })\mathit{\gamma }({\zeta }_2)+(\mathit{\lambda }-{\zeta }_1)\mathit{\gamma }({\zeta }_1)+\mathit{\gamma }(\mathit{\lambda })({\zeta }_2-{\zeta }_1)}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2)\left|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_1)\right|\mathit{d}\mathrm{\ell }$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\int }_0^1(\mathrm{\ell }-{\mathrm{\ell }}^2)\left|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_2)\right|\mathit{d}\mathrm{\ell }$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left({\int }_0^1\mathrm{\ell }\right)}^{1-\frac{1}{{\beta }_2}}{\left({\int }_0^1\mathrm{\ell }{(1-\mathrm{\ell })}^{{\beta }_2}{\left|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_1)\right|}^{{\beta }_2}\mathit{d}\mathrm{\ell }\right)}^{\frac{1}{{\beta }_2}}$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left({\int }_0^1\mathrm{\ell }\right)}^{1-\frac{1}{{\beta }_2}}{\left({\int }_0^1\mathrm{\ell }{(1-\mathrm{\ell })}^{{\beta }_2}{\left|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathrm{\ell }\mathit{\lambda }+(1-\mathrm{\ell }){\zeta }_2)\right|}^{{\beta }_2}\mathit{d}\mathrm{\ell }\right)}^{\frac{1}{{\beta }_2}}.$

By applying $\mathit{h}$-convexity of $|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}|$, we get

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left({\int }_0^1\mathrm{\ell }\right)}^{1-\frac{1}{{\beta }_2}}{\left({\int }_0^1\mathrm{\ell }{(1-\mathrm{\ell })}^{{\beta }_2}(\mathit{h}(\mathrm{\ell })|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{h}(1-\mathrm{\ell })|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1{)|}^{{\beta }_2}\mathit{d}\mathrm{\ell }\right)}^{\frac{1}{{\beta }_2}}$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left({\int }_0^1\mathrm{\ell }\right)}^{1-\frac{1}{{\beta }_2}}{\left({\int }_0^1\mathrm{\ell }{(1-\mathrm{\ell })}^{{\beta }_2}(\mathit{h}(\mathrm{\ell })|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{h}(1-\mathrm{\ell })|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1{)|}^{{\beta }_2}\mathit{d}\mathrm{\ell }\right)}^{\frac{1}{{\beta }_2}}$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left({\int }_0^1\mathrm{\ell }\right)}^{1-\frac{1}{{\beta }_2}}{\left({\int }_0^1\mathrm{\ell }{(1-\mathrm{\ell })}^{{\beta }_2}(\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1{)|}^{{\beta }_2}\mathit{d}\mathrm{\ell }\right)}^{\frac{1}{{\beta }_2}}$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left({\int }_0^1\mathrm{\ell }\right)}^{1-\frac{1}{{\beta }_2}}{\left({\int }_0^1\mathrm{\ell }{(1-\mathrm{\ell })}^{{\beta }_2}(\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1{)|}^{{\beta }_2}\mathit{d}\mathrm{\ell }\right)}^{\frac{1}{{\beta }_2}}$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(B(2,{\beta }_2+1)(\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1{)|}^{{\beta }_2})\right)}^{\frac{1}{{\beta }_2}}$

$+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(B(2,{\beta }_2+1)(\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(\mathit{\lambda }{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2{)|}^{{\beta }_2})\right)}^{\frac{1}{{\beta }_2}}.$

Hence, the proof is done.

Example 5.

To validate the inequality presented in Theorem 9, one may inspect the graph of inequality (9). For this analysis, we substitute $\mathit{\gamma }(\mathrm{\ell })=e^{\mathrm{\ell }}$ and obtain the following result.

$-\left[\left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(B(2,{\beta }_2+1)(\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_1}{|}^{{\beta }_2})\right)}^{\frac{1}{{\beta }_2}}\right.$

$\left.+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(B(2,{\beta }_2+1)(\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_2}{|}^{{\beta }_2})\right)}^{\frac{1}{{\beta }_2}}\right]$

$\le \frac{({\zeta }_2-\mathit{\lambda })e^{{\zeta }_2}+(\mathit{\lambda }-{\zeta }_1)e^{{\zeta }_1}+({\zeta }_2-{\zeta }_1)e^{\lambda }}{2({\zeta }_2-{\zeta }_1)}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}{e}^{\mathrm{\ell }}\mathit{d}\mathrm{\ell }$

$\le \left|\frac{{(\lambda -{\zeta }_1)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(B(2,{\beta }_2+1)(\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_1}{|}^{{\beta }_2})\right)}^{\frac{1}{{\beta }_2}}$

(10) $+\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(B(2,{\beta }_2+1)(\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_2}{|}^{{\beta }_2})\right)}^{\frac{1}{{\beta }_2}}.$(10)

Case (i) By setting the parameters $\mathit{m}=1$ and $\mathit{\lambda }=1$ with $\mathit{L}({\zeta }_1,{\zeta }_2)$, $\mathit{M}({\zeta }_1,{\zeta }_2)$ and $\mathit{R}({\zeta }_1,{\zeta }_2)$ denoting the left, middle and right components and considering the ranges of ${\zeta }_1\in [1,10]$ and ${\zeta }_2\in [11,20]$, the graph of the double inequality (10) can be presented as follows.

Case (ii) By setting the parameters as $\mathit{m}=1$, ${\zeta }_1=1$ and ${\zeta }_2=2$ and specifying the functions $\mathit{L}(\mathit{\lambda })$, $\mathit{M}(\mathit{\lambda })$ and $\mathit{R}(\mathit{\lambda })$ as the left, middle and right components and while examining the range of $\mathit{\lambda }\in [10,20]$, the graph of the double inequality (10) can be presented as follows.

The Figures 9,10 visually confirm the results obtained in 3D and 2D graphs of Equation (8)(8) $+\left.\left|\frac{{(\lambda -{\zeta }_2)}^3}{2({\zeta }_2-{\zeta }_1)}\right|{\left(\frac{1}{{\beta }_1+1}\right)}^{\frac{1}{{\beta }_1}}{\left(\frac{\mathit{m}|e^{\lambda }{|}^{{\beta }_2}+\mathit{m}|e^{{\zeta }_2}{|}^{{\beta }_2}}{{\beta }_2+1}\right)}^{\frac{1}{{\beta }_2}}\right].$(8) .

Figure 9. The graph that show the inequality (10) corresponds to the above mentioned parameters.

Figure 10. The graph that show the inequality (10) corresponds to the above mentioned parameters.

4. Some application to main results in term of means

The concept of the ‘mean’ is widely used in various fields and contexts to summarize and analyse data. The mean is a measure of central tendency that represents the average or typical value in a data set. The purpose of this section is to illustrate practical uses of key results related to means.

The means are expressed as follows:

(i) The arithmetic means

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

(ii) The logarithmic mean

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

Proposition 1.

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

$\left|\mathit{A}(e^{{\zeta }_1},e^{{\zeta }_2})-\mathit{L}(e^{{\zeta }_1},e^{{\zeta }_2})\right|\le \frac{{({\zeta }_1-{\zeta }_2)}^2\left(e^{{\zeta }_1}+e^{{\zeta }_2}\right)}{12}.$

Proof.

By using Theorem 5 and substituting $\mathit{\lambda }={\zeta }_1$ in (1), we have

(11) $\left|\frac{\gamma ({\zeta }_1)+\mathit{\gamma }({\zeta }_2)}{2}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|\le \frac{m{({\zeta }_1-{\zeta }_2)}^2\left(|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1)|+|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2)|\right)}{12}.$(11)

By substituting $\mathit{\gamma }(\mathrm{\ell })=e^{\mathrm{\ell }}$ and $\mathit{m}=1$ in (11), we can write as

$\left|\frac{e^{{\zeta }_1}+e^{{\zeta }_2}}{2}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}{e}^{\mathrm{\ell }}\mathit{d}\mathrm{\ell }\right|\le \frac{{({\zeta }_1-{\zeta }_2)}^2\left(e^{{\zeta }_1}+e^{{\zeta }_2}\right)}{12}$

$\left|\frac{e^{{\zeta }_1}+e^{{\zeta }_2}}{2}-\frac{e^{{\zeta }_2}-e^{{\zeta }_1}}{{\zeta }_2-{\zeta }_1}\right|\le \frac{{({\zeta }_1-{\zeta }_2)}^2\left(e^{{\zeta }_1}+e^{{\zeta }_2}\right)}{12}.$

Proposition 2

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

$\left|\mathit{A}(e^{{\zeta }_1},e^{{\zeta }_2})-\mathit{L}(e^{{\zeta }_1},e^{{\zeta }_2})\right|\le \frac{{({\zeta }_2-{\zeta }_1)}^2{\left(e^{2{\zeta }_1}+e^{2{\zeta }_2}\right)}^{\frac{1}{2}}}{12}.$

Proof.

By using Theorem 6 and substituting $\mathit{\lambda }={\zeta }_1$ in (3), we have

$\left|\frac{\gamma ({\zeta }_1)+\mathit{\gamma }({\zeta }_2)}{2}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

(12) $\le \frac{{({\zeta }_2-{\zeta }_1)}^2}{2}{\left(\frac{1}{6}\right)}^{1-\frac{1}{{\beta }_2}}{\left(\frac{\mathit{m}|{\mathit{\gamma }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1{)|}^{{\beta }_2}+\mathit{m}|{\gamma }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}{({\zeta }_2)}^{{\beta }_2}|}{6}\right)}^{\frac{1}{{\beta }_2}}.$(12)

By substituting $\mathit{\gamma }(\mathrm{\ell })=e^{\mathrm{\ell }}$, $\mathit{m}=1$ and ${\beta }_2=2$ in (12), we can write as

$\left|\frac{e^{{\zeta }_1}+e^{{\zeta }_2}}{2}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}{e}^{\mathrm{\ell }}\mathit{d}\mathrm{\ell }\right|\le \frac{{({\zeta }_2-{\zeta }_1)}^2{\left(e^{2{\zeta }_1}+e^{2{\zeta }_2}\right)}^{\frac{1}{2}}}{12}$

$\left|\frac{e^{{\zeta }_1}+e^{{\zeta }_2}}{2}-\frac{e^{{\zeta }_2}-e^{{\zeta }_1}}{{\zeta }_2-{\zeta }_1}\right|\le \frac{{({\zeta }_2-{\zeta }_1)}^2{\left(e^{2{\zeta }_1}+e^{2{\zeta }_2}\right)}^{\frac{1}{2}}}{12}.$

Proposition 3.

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

$\left|\mathit{A}(e^{{\zeta }_1},e^{{\zeta }_2})-\mathit{L}(e^{{\zeta }_1},e^{{\zeta }_2})\right|\le \frac{0.775759({\zeta }_1-{\zeta }_2{)}^3{(e^{2{\zeta }_1}+e^{2{\zeta }_2})}^{\frac{1}{2}}}{2}.$

Proof.

By using Theorem 7 and substituting $\mathit{\lambda }={\zeta }_1$ in (5), we have

$\left|\frac{\gamma ({\zeta }_1)+\mathit{\gamma }({\zeta }_2)}{2}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}\mathit{\gamma }(\mathrm{\ell })\mathit{d}\mathrm{\ell }\right|$

(13) $\le \frac{{({\zeta }_1-{\zeta }_2)}^3}{2}{\left(\frac{\mathrm{\Gamma }({\beta }_1+1)}{\mathrm{\Gamma }({\beta }_1+\frac{3}{2})}\right)}^{\frac{1}{{\beta }_1}}{\left(|{\mathit{ }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_1{)|}^{{\beta }_2}+|{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}({\zeta }_2{)|}^{{\beta }_2}\right)}^{\frac{1}{{\beta }_2}}.$(13)

By substituting $\mathit{\gamma }(\mathrm{\ell })=e^{\mathrm{\ell }}$, $\mathit{m}=1$ and ${\beta }_1={\beta }_2=2$ in (13), we can write as

$\left|\frac{e^{{\zeta }_1}+e^{{\zeta }_2}}{2}-\frac{1}{{\zeta }_2-{\zeta }_1}{\int }_{{\zeta }_1}^{{\zeta }_2}{e}^{\mathrm{\ell }}\mathit{d}\mathrm{\ell }\right|\le \frac{0.775759({\zeta }_1-{\zeta }_2{)}^3{(e^{2{\zeta }_1}+e^{2{\zeta }_2})}^{\frac{1}{2}}}{2}$

$\left|\frac{e^{{\zeta }_1}+e^{{\zeta }_2}}{2}-\frac{e^{{\zeta }_2}-e^{{\zeta }_1}}{{\zeta }_2-{\zeta }_1}\right|\le \frac{0.775759({\zeta }_1-{\zeta }_2{)}^3{(e^{2{\zeta }_1}+e^{2{\zeta }_2})}^{\frac{1}{2}}}{2}.$

5. Conclusions

Hermite-Hadamard inequalities are a notable set of mathematical results that establish connections between the integral and average value of a real-valued function over a closed interval. These inequalities provide valuable insights into the behaviour of functions and play a crucial role in various branches of mathematics, including calculus, real analysis and convex functions. They have practical applications in diverse areas such as physics, economics, optimization and engineering. In this paper, we have established some novel fractional integral inequalities of Hermite-Hadamard-type for a class of differentiable trigonometrically-convex functions. Several novel estimates of Hermite-Hadamard inequality are produced. In addition, some limiting cases are given as corollaries. Further, new inequalities involving special means, such as arithmetic, geometric, logarithmic and some other well-known means, are generated as a result of some of our main findings. This article is supposed to constructively contribute and applicable to the current literature.

Authors’ contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgement

The author Thabet Abdeljawad would like to thank Prince Sultan University through the TAS research lab.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research did not receive any specific funding.