On Fractional Integral Inequalities of Riemann Type for Composite Convex Functions and Applications

Abstract

In this study, we apply fractional calculus on certain convex functions and derive many novel mean-type inequalities by employing fractional calculus and convexity theory. In order to investigate fractional mean inequalities, we first build an identity in this study. Then, with its help, we derive many mean-type inequalities and estimate the error of HH inequality using a generalized version of RL-fractional integrals and certain classes of convex functions. The results obtained are validated by taking specific functions. Many mean-type inequalities that exist in the literature are generalized by the main results of this study.

1. Introduction

In recent times, fractional calculus has gained much attention due to fractional calculus having received a lot of attention recently as a result of its widespread use in contemporary sciences. Numerous real-world issues have been resolved using fractional theory, including estimating blood alcohol level [1], modelling in continuum mechanics [2], controlling hexapod robots [3], determining non-local continua [4], administrating drug dosages [5], forming memory through membranes [6], determining semilunar heat valve vibrations [7], diffusing water in sand [8], determining the growth rate and population of poisonous bacteria [9], analyzing social and economic cycles [10], and many others [11,12,13].

Mathematical inequalities span many fields of science. Their importance has increased greatly due to their widespread use, especially in estimating the solutions of bounded value problems. Inequalities and convexity are very closely related. An important inequality that describes and reveals this convexity is known as the Hermite Hadamard (HH) inequality [14], which is given as

Theorem 1.

Let $\xi :U\subseteq R\to R$ be a convex function, then the following inequality holds.

$$\begin{array}{c}\hfill \xi \left({\displaystyle \frac{p_1+p_2}{2}}\right)\le {\displaystyle \frac{1}{p_2-p_1}}{\int }_{p_1}^{p_2}\xi \left(\delta \right)d\delta \le {\displaystyle \frac{\xi \left(p_1\right)+\xi \left(p_2\right)}{2}}.\end{array}$$

The HH inequality is formulated for different interesting situations such as those in [15,16] for generalized fractional operators and those in [17,18,19] via different convexities. The convexities in certain types of inequalities can also be seen in [20,21].

Fractional operators provide more generalized and advance results even where ordinary calculus fails by reducing many constraints. Their growing use has attracted many scientists towards this non-integer order calculus. Farid et al. [22] applied RL fractional integrals to study the HH inequality with convexity for monotonic functions.

To generalize and extend the results, we apply the $(k,s)$-Riemann integral [23] and find a special identity that helps us to obtain many mean-type inequalities already known from the literature. We also estimate the error of mean-type inequalities. The main results of our study are more advanced and generalized from the pre-existing literature. Moreover, the results of [24,25,26] are connected with that from our work.

This paper is organize in the following way. Section 2 states some important related preliminaries. Section 3 consists of a generalized identity and the theorems based on this established identify generalize the estimates presented in [22]. In Section 4, some conclusions are presented.

2. Preliminaries

Here, we present some basic facts that are needed to understand the main results.

Definition 1.

A function $\xi :[m,n]\to (-\infty ,\infty )$ is a convex function if it satisfies the inequality;

$$\begin{array}{c}\xi (\delta x+(1-\delta \left)y\right)\le \delta \xi \left(x\right)+(1-\delta )\xi \left(y\right),\hfill \end{array}$$

where $0\le \delta \le 1$ and $x,y\in [m,n]$.

The next definition describes the criteria for a certain function to be convex with respect to some other function with a strictly monotone property.

Definition 2

([27]). The function ξ is convex with respect to strictly the monotone function ϱ if $(\xi \circ {\varrho }^{-1})$ is a convex function.

One can read many different aspects of the HH inequality and its generalizations in [28,29,30]. A lot of interesting applications have encouraged mathematicians to investigate its different versions. The HH inequality for a function satisfying Definition 2 can be seen in the next theorem.

Theorem 2

([31]). Suppose that $I_1,I_2\subset (-\infty ,+\infty )$, $\varrho :[m,n]\subset I_2\to (-\infty ,\infty )$ is a strictly monotone function and $\xi :[m,n]\subset I_1\to (-\infty ,\infty )$ is convex with respect to ϱ. Then,

$$\begin{array}{c}\hfill \xi \left({\varrho }^{-1}\left({\displaystyle \frac{\varrho \left(m\right)+\varrho \left(n\right)}{2}}\right)\right)\le {\displaystyle \frac{1}{\varrho \left(n\right)-\varrho \left(m\right)}}{\int }_{\varrho \left(m\right)}^{\varrho \left(n\right)}\xi \left({\varrho }^{-1}\left(\delta \right)\right)d\delta \le {\displaystyle \frac{\xi \left(m\right)+\xi \left(n\right)}{2}}.\end{array}$$

The classical gamma function is extended in the next definition.

Definition 3

([32]). The k-Gamma function symbolized as ${\Gamma }_k$ is defined as

${\Gamma }_k\left(w\right)={\int }_0^{\infty }{e}^{-\frac{{\delta }^k}{k}}{\delta }^{w-1}d\delta$ where $w\in \mathbb{C}$ and $Re\left(w\right)>0$.

The k-Gamma function has a similar property to that of the classical gamma function i.e., $\delta {\Gamma }_k\left(\delta \right)={\Gamma }_k(\delta +k)$.

The Riemann–Liouville fractional integrals (which we simply call RL-fractional integrals) are defined as follows:

Definition 4

([33]). The left-sided and right-sided RL-fractional integrals of f having an order $\beta >0$ are described as

$$\begin{array}{cccc}\hfill & I_{m^{+}}^{\beta }f\left(x\right)=\frac{1}{\Gamma \left(\beta \right)}{\int }_m^x{(x-\delta )}^{\beta -1}f\left(\delta \right)d\delta ,\hfill & \hfill & x>m,\hfill \\ \hfill & I_{n^{-}}^{\beta }f\left(x\right)=\frac{1}{\Gamma \left(\beta \right)}{\int }_x^n{(\delta -x)}^{\beta -1}f\left(\delta \right)d\delta ,\hfill & \hfill & x<n.\hfill \end{array}$$

The next definition presents one of the extended versions of the RL-fractional integral given in [34].

Definition 5.

The left- and right-sided (k,s)-RL-fractional integrals of a function f of the order $\beta >0$ are defined as follows:

$$\begin{array}{ccc}\hfill {}_k^s{I}_{m^{+}}^{\beta }f\left(x\right)={\displaystyle \frac{{(s+1)}^{1-\frac{\beta }{k}}}{k{\Gamma }_k\left(\beta \right)}}{\int }_m^x{\left(x^{s+1}-{\delta }^{s+1}\right)}^{\frac{\beta }{k}-1}{\delta }^{s}f\left(\delta \right)d\delta ,& \hfill & \hfill x>m,\end{array}$$

$$\begin{array}{ccc}\hfill {}_k^s{I}_{n^{-}}^{\beta }f\left(x\right)={\displaystyle \frac{{(s+1)}^{1-\frac{\beta }{k}}}{k{\Gamma }_k\left(\beta \right)}}{\int }_x^n{\left({\delta }^{s+1}-x^{s+1}\right)}^{\frac{\beta }{k}-1}{\delta }^{s}f\left(\delta \right)d\delta ,& \hfill & \hfill x<n.\end{array}$$

3. Main Results

In this section, we present many mean-type inequalities obtained by applying the extended version of the RL-fractional integral “$(k,s)$-RL-fractional integral” and certain classes of the convex functions.

The basic identity essential for our main results is obtained in the following lemma, which is further used in estimating the error of the HH inequality.

Lemma 1.

Let the two real mappings ξ and ϱ be defined on $[m,n]$ with $m<n$ such that ϱ has the strictly monotone property. If $\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}$ is differentiable and ${(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1})}^{\prime }\in L[m,n]$, then the following equation holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\xi \left(m\right)+\xi \left(n\right)}{2}}-{\displaystyle \frac{{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{2{\left({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)\right)}^{\frac{u}{k}}}}\left({}_k^s{I}_{{{\varrho }^{s+1}\left(m\right)}^{+}}^u\xi \left(n\right){+}_k^s{I}_{{{\varrho }^{s+1}\left(n\right)}^{-}}^u\xi \left(m\right)\right)\hfill \\ \hfill & ={\displaystyle \frac{{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)}{2}}{\int }_0^1\left({(1-\delta )}^{\frac{u}{k}}-{\delta }^{\frac{u}{k}}\right){\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left(\delta {\varrho }^{s+1}\left(m\right)+(1-\delta ){\varrho }^{s+1}\left(n\right)\right)d\delta .\hfill \end{array}$$

(1)

Proof. 

By evaluating the integral, we have

$$\begin{array}{cc}\hfill & {\int }_0^1{\left(1-\delta \right)}^{\frac{u}{k}}{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left(\delta {\varrho }^{s+1}\left(m\right)+\left(1-\delta \right){\varrho }^{s+1}\left(n\right)\right)d\delta \hfill \\ \hfill & =\frac{{(1-\delta )}^{\frac{u}{k}}\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)\left(\delta {\varrho }^{s+1}\left(m\right)+(1-\delta ){\varrho }^{s+1}\left(n\right)\right)}{{\varrho }^{s+1}\left(m\right)-{\varrho }^{s+1}\left(n\right)}{|}_o^1\hfill \\ \hfill & +\frac{u}{k}{\int }_0^1\frac{{(1-\delta )}^{\frac{u}{k}-1}\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)(\delta {\varrho }^{s+1}\left(m\right)+(1-\delta ){\varrho }^{s+1}\left(n\right))d\delta }{{\varrho }^{s+1}\left(m\right)-{\varrho }^{s+1}\left(n\right)}\hfill \\ \hfill & =\frac{\xi \left(n\right)}{{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)}-\frac{\frac{u}{k}}{{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)}\hfill \\ \hfill & \times {\int }_0^1{\left(1-\delta \right)}^{\frac{u}{k}-1}\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)\left(\delta {\varrho }^{s+1}\left(m\right)+\left(1-\delta \right){\varrho }^{s+1}\left(n\right)\right)d\delta \hfill \\ \hfill & =\frac{\xi \left(n\right)}{{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)}-\frac{\frac{u}{k}}{{\left({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)\right)}^{\frac{u}{k}+1}}\hfill \\ \hfill & \times {\int }_{{\varrho }^{s+1}\left(m\right)}^{{\varrho }^{s+1}\left(n\right)}{\left(z^{s+1}-{\varrho }^{s+1}\left(m\right)\right)}^{\frac{u}{k}-1}\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)\left(z^{s+1}\right)z^s(s+1)dz\hfill \\ \hfill & =\left(\frac{\xi \left(n\right)}{{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)}\right)-\frac{{\left(s+1\right)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{{\left({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)\right)}^{\frac{u}{k}+1}}\left({}_k^s{I}_{{{\varrho }^{s+1}\left(n\right)}^{-}}^u\xi \left(m\right)\right).\hfill \end{array}$$

(2)

Similarly, integrating by parts, we obtain

$$\begin{array}{cc}\hfill & {\int }_0^1{\delta }^{\frac{u}{k}}{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left(\delta {\varrho }^{s+1}\left(m\right)+\left(1-\delta \right){\varrho }^{s+1}\left(n\right)\right)d\delta \hfill \\ \hfill & =\frac{-\xi \left(n\right)}{{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)}+\frac{{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{{\left({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)\right)}^{\frac{u}{k}+1}}\left({}_k^s{I}_{{{\varrho }^{s+1}\left(m\right)}^{+}}^u\xi \left(n\right)\right).\hfill \end{array}$$

(3)

Substituting (2) and (3) in (1) gives the desired result. □

Lemma 1 helps us to derive the following error estimation of the HH inequality given in Theorem 2.

Theorem 3.

Let the two real mappings ξ and ϱ be defined on $[m,n]$ with $m<n$ such that ϱ has the strictly monotone property. If $\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}$ is differentiable and ${(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1})}^{\prime }\in L[m,n]$, then the inequality

$$\begin{array}{cc}\hfill & |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{{\left(s+1\right)}^{\frac{u}{k}}{\Gamma }_k\left(u+k\right)}{2{\left({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)\right)}^{\frac{u}{k}}}\left({}_k^s{I}_{{{\varrho }^{s+1}\left(m\right)}^{+}}^u\xi \left(n\right){+}_k^s{I}_{{{\varrho }^{s+1}\left(n\right)}^{-}}^u\xi \left(m\right)\right)|\hfill \\ \hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{2\left(\frac{u}{k}+1\right)}\left(1-\frac{1}{2^{\frac{u}{k}}}\right)\left(\right|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left({\varrho }^{s+1}\left(m\right)\right)|+|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left({\varrho }^{s+1}\left(n\right)\right)\left|\right),\hfill \end{array}$$

(4)

holds when $|{(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1})}^{\prime }|$ is convex.

Proof. 

By utilizing Lemma 1 with simple calculus, the above inequality can be described by

$$\begin{array}{cc}\hfill & |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{2{\left({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)\right)}^{\frac{u}{k}}}\left({}_k^s{I}_{{{\varrho }^{s+1}\left(m\right)}^{+}}^u\xi \left(n\right){+}_k^s{I}_{{{\varrho }^{s+1}\left(n\right)}^{-}}^u\xi \left(m\right)\right)|\hfill \\ \hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{2}{\int }_0^1|{\left(1-\delta \right)}^{\frac{u}{k}}-{\delta }^{\frac{u}{k}}||{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left(\delta {\varrho }^{s+1}\left(m\right)+\left(1-\delta \right){\varrho }^{s+1}\left(n\right)\right)|d\delta .\hfill \end{array}$$

(5)

Since ${\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }$ is convex, using this fact on the right-hand side of (5) will imply the following:

$$\begin{array}{cc}\hfill & |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{2{\left({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)\right)}^{\frac{u}{k}}}\left({}_k^s{I}_{{{\varrho }^{s+1}\left(m\right)}^{+}}^u\xi \left(n\right){+}_k^s{I}_{{{\varrho }^{s+1}\left(n\right)}^{-}}^u\xi \left(m\right)\right)|\hfill \\ \hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{2}{\int }_0^1|{\left(1-\delta \right)}^{\frac{u}{k}}-{\delta }^{\frac{u}{k}}|\hfill \\ \hfill & \times \left(\delta \right|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right)|+\left(1-\delta \right)|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right)|)d\delta \hfill \\ \hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{2}{\int }_0^{\frac{1}{2}}|{\left(1-\delta \right)}^{\frac{u}{k}}-{\delta }^{\frac{u}{k}}||{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left(\delta {\varrho }^{s+1}\left(m\right)+\left(1-\delta \right){\varrho }^{s+1}\left(n\right)\right)|d\delta \hfill \\ \hfill & +{\int }_{\frac{1}{2}}^1|{\left(1-\delta \right)}^{\frac{u}{k}}-{\delta }^{\frac{u}{k}}||{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left(\delta {\varrho }^{s+1}\left(m\right)+\left(1-\delta \right){\varrho }^{s+1}\left(n\right)\right)|d\delta \hfill \\ \hfill & =\frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{2}[|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right)|{\int }_0^{\frac{1}{2}}\left(\delta {\left(1-\delta \right)}^{\frac{u}{k}}-{\delta }^{\frac{u}{k}+1}\right)d\delta \hfill \\ \hfill & +\left|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right)\right|{\int }_0^{\frac{1}{2}}\left({\left(1-\delta \right)}^{\frac{u}{k}+1}-{\delta }^{\frac{u}{k}}\left(1-\delta \right)\right)d\delta \hfill \\ \hfill & +|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right)|{\int }_{\frac{1}{2}}^1\left({\delta }^{\frac{u}{k}+1}-\delta {\left(1-\delta \right)}^{\frac{u}{k}}\right)d\delta \hfill \\ \hfill & +\left|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right)\right|{\int }_{\frac{1}{2}}^1({\delta }^{\frac{u}{k}}\left(1-\delta \right)-{\left(1-\delta \right)}^{\frac{u}{k}+1}d\delta \left)\right].\hfill \end{array}$$

Thus, we can obtain our desired result after a few computations. □

In the next examples, we discuss the applications of Theorem 3 for some specific functions.

Example 1.

By substituting ${\varrho }^{s+1}\left(x\right)=e^x$ in (4), we have the inequality

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{\xi \left(m\right)+\xi \left(n\right)}{2}}-\frac{{(s+1)}^{\frac{u}{k}}{\Gamma }_k\left(u+k\right)}{2{\left(e^n-e^m\right)}^{\frac{u}{k}}}\left({}_k^s{I}_{{e^m}^{+}}^u\xi \left(n\right)+{}_k^s{I}_{{e^n}^{-}}^u\xi \left(m\right)\right)|\hfill \\ \hfill & \le \frac{|e^n-e^m|}{2(\frac{u}{k}+1)}\left(1-\frac{1}{2^{\frac{u}{k}}}\right)\left(e^{-m}\left|{\xi }^{\prime }\left(m\right)\right|+e^{-n}\left|{\xi }^{\prime }\left(n\right)\right|\right).\hfill \end{array}$$

Example 2.

By substituting ${\varrho }^{s+1}\left(x\right)=e^x$ and ${\displaystyle \frac{u}{k}}=1$ in (4), we have the inequality

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{\xi \left(m\right)+\xi \left(n\right)}{2}}-\frac{k(s+1)}{2\left(e^n-e^m\right)}\left({}_k^s{I}_{{e^m}^{+}}^u\xi \left(n\right)+{}_k^s{I}_{{e^n}^{-}}^u\xi \left(m\right)\right)|\hfill \\ \hfill & \le \frac{|e^n-e^m|}{8}\left(e^{-m}\left|{\xi }^{\prime }\left(m\right)\right|+e^{-n}\left|{\xi }^{\prime }\left(n\right)\right|\right).\hfill \end{array}$$

Example 3.

By substituting ${\varrho }^{s+1}\left(x\right)={\displaystyle \frac{1}{x}}$ in (4), we obtain the following:

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{\xi \left(m\right)+\xi \left(n\right)}{2}}-\frac{{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{2}{\left({\displaystyle \frac{mn}{n-m}}\right)}^{\frac{u}{k}}\left({}_k^s{I}_{{\frac{1}{m}}^{-}}^u\xi \circ g\left({\displaystyle \frac{1}{n}})+{{}_k^s{I}^u}_{{\frac{1}{n}}^{+}}\xi \circ g({\displaystyle \frac{1}{m}}\right)\right)|\hfill \\ \hfill & \le {\displaystyle \frac{|m-n|}{2\left|mn\right|\left(\frac{u}{k}+1\right)}}\left(1-\frac{1}{2^{\frac{u}{k}}}\right)\left(m^2\left|{\xi }^{\prime }\left(m\right)\right|+n^2\left|{\xi }^{\prime }\left(n\right)\right|\right),\hfill \end{array}$$

where $g\left(\delta \right)={\displaystyle \frac{1}{\delta }}$.

Example 4.

By substituting ${\varrho }^{s+1}\left(x\right)={\displaystyle \frac{1}{x}}$ and ${\displaystyle \frac{u}{k}}=1$ in (4), we obtain

$$\begin{array}{c}\hfill |{\displaystyle \frac{\xi \left(m\right)+\xi \left(n\right)}{2}}-{\displaystyle \frac{k(s+1)mn}{n-m}}{\int }_{\frac{1}{n}}^{\frac{1}{m}}\left(\xi \circ g\right)\left(\delta \right)d\delta |\le {\displaystyle \frac{|m-n|}{8\left|mn\right|}}\left(m^2\left|{\xi }^{\prime }\left(m\right)\right|+n^2|{\xi }^{\prime }\left(n\right)|\right),\end{array}$$

where $g\left(\delta \right)={\displaystyle \frac{1}{\delta }}$.

Example 5.

By substituting ${\varrho }^{s+1}\left(x\right)=x^r$, $r\ne 0$ in (4), we obtain the inequality

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{\xi \left(m\right)+\xi \left(n\right)}{2}}-\frac{r^{\frac{u}{k}}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{2{(n^r-m^r)}^{\frac{u}{k}}}\left({}_k^{r,s}{I}_{m^{+}}^u\xi \left(\delta \right)+{{}_k^{r,s}I}_{n^{-}}^u\xi \left(\delta \right)\right)\hfill \\ \hfill & \le \frac{|n^r-m^r|}{2\left|r\right|(\frac{u}{k}+1)}\left(1-\frac{1}{2^{\frac{u}{k}}}\right)\left(m^{1-r}\left|{\xi }^{\prime }\left(m\right)\right|+n^{1-r}\left|{\xi }^{\prime }\left(n\right)\right|\right).\hfill \end{array}$$

Example 6.

By substituting ${\varrho }^{s+1}\left(x\right)=x^r$, $r\ne 0$ and ${\displaystyle \frac{u}{k}}=1$ in (4), we obtain

$$\begin{array}{c}\hfill |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{k(s+1)\left(r\right)}{2(n^r-m^r)}{\int }_n^m{\delta }^{r-1}f\left(\delta \right)d\delta |\le \frac{|n^r-m^r|}{8\left|r\right|}\left(m^{1-r}\left|{\xi }^{\prime }\left(m\right)\right|+n^{1-r}\left|{\xi }^{\prime }\left(n\right)\right|\right).\end{array}$$

Example 7.

By substituting ${\varrho }^{s+1}\left(x\right)=ln\left(x\right)$ in (4), we have the inequality

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{\xi \left(m\right)+\xi \left(n\right)}{2}}-\frac{{(s+1)}^{\frac{u}{k}}{\Gamma }_k\left(u+k\right)}{2{\left(ln\left(n\right)-ln\left(m\right)\right)}^{\frac{u}{k}}}\left({}_k^s{I}_{ln{\left(m\right)}^{+}}^u\xi \left(n\right)+{}_k^s{I}_{ln{\left(n\right)}^{-}}^u\xi \left(m\right)\right)|\hfill \\ \hfill & \le \frac{|ln(n)-ln(m\left)\right|}{2(\frac{u}{k}+1)}\left(1-\frac{1}{2^{\frac{u}{k}}}\right)\left(m\left|{\xi }^{\prime }\left(m\right)\right|+n\left|{\xi }^{\prime }\left(n\right)\right|\right).\hfill \end{array}$$

Example 8.

By substituting ${\varrho }^{s+1}\left(x\right)=ln\left(x\right)$ and ${\displaystyle \frac{u}{k}}=1$ in (4), we obtain

$$\begin{array}{c}\hfill |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{k(s+1)}{ln\left(n\right)-ln\left(m\right)}{\int }_m^n\frac{\xi \left(\delta \right)}{\delta }d\delta |\le {\displaystyle \frac{ln\left(n\right)-ln\left(m\right)}{8}}\left(m\left|{\xi }^{\prime }\left(m\right)\right|+n\left|{\xi }^{\prime }\left(n\right)\right|\right).\end{array}$$

Theorem 4.

Let the two real mappings ξ and ϱ be defined on $[m,n]$ with $m<n$ such that ϱ has the strictly monotone property. If $\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}$ is differentiable and ${(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1})}^{\prime }\in L[m,n]$, then the inequality

$$\begin{array}{cc}\hfill & |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{{\left(s+1\right)}^{\frac{u}{k}}{\Gamma }_k\left(u+k\right)}{2{({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right))}^{\frac{u}{k}}}\left({}_k^s{I}_{{{\varrho }^{s+1}\left(m\right)}^{+}}^u\xi \left(n\right){+}_k^s{I}_{{{\varrho }^{s+1}\left(n\right)}^{-}}^u\xi \left(m\right)\right)|\hfill \\ \hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{2^{\frac{1}{q}}(\frac{u}{k}+1)}\left(1-\frac{1}{2^{\frac{u}{k}}}\right)\hfill \\ \hfill & \times {\left(|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left({\varrho }^{s+1}\left(m\right)\right){|}^q+|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left({\varrho }^{s+1}\left(n\right)\right){|}^q\right)}^{\frac{1}{q}}\hfill \end{array}$$

(6)

holds whenever $|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{|}^q$ is convex for $q\ge 1$.

Proof. 

If $q=1$, then the above inequality can be obtained by using basic calculus in Lemma 1 and $|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{|}^q$ is convex.

Let $q>1$. The right-hand side of Lemma 1, along with the power mean inequality and basic calculus, then implies the following inequality:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{2{({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right))}^{\frac{u}{k}}}\left({}_k^s{I}_{{{\varrho }^{s+1}\left(m\right)}^{+}}^u\xi \left(n\right){+}_k^s{I}_{{{\varrho }^{s+1}\left(n\right)}^{-}}^u\xi \left(m\right)\right)|\hfill \\ \hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{2}\hfill \\ \hfill & \times \left({\int }_0^1|{\left(1-\delta \right)}^{\frac{u}{k}}-{\delta }^{\frac{u}{k}}|{)}^{1-\frac{1}{q}}({\int }_0^1\right|{\left(1-\delta \right)}^{\frac{u}{k}}-{\delta }^{\frac{u}{k}}|\hfill \\ \hfill & \times |{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left(\delta {\varrho }^{s+1}\left(m\right)+\left(1-\delta \right){\varrho }^{s+1}\left(n\right)\right){|}^{q}d\delta {)}^{\frac{1}{q}}.\hfill \end{array}\end{array}$$

(7)

Consider

$$\begin{array}{cc}\hfill & {\int }_0^1|{\left(1-\delta \right)}^{\frac{u}{k}}-{\delta }^{\frac{u}{k}}|d\delta ={\int }_0^{\frac{1}{2}}\left({\left(1-\delta \right)}^{\frac{u}{k}}-{\delta }^{\frac{u}{k}}d\delta \right)+{\int }_{\frac{1}{2}}^1\left({\delta }^{\frac{u}{k}}-{\left(1-\delta \right)}^{\frac{u}{k}}d\delta \right)\hfill \\ \hfill & =\frac{2}{(\frac{u}{k}+1)}\left(1-\frac{1}{2^{\frac{u}{k}}}\right).\hfill \end{array}$$

(8)

Since $|{(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1})}^{\prime }{|}^q$ is convex,

$$\begin{array}{cc}\hfill & {\int }_0^1|{\left(1-\delta \right)}^{\frac{u}{k}}-{\delta }^{\frac{u}{k}}||{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left(\delta {\varrho }^{s+1}\left(m\right)+(1-\delta ){\varrho }^{s+1}\left(n\right)\right){|}^{q}d\delta \hfill \\ \hfill & \le {\int }_0^{\frac{1}{2}}\left({\left(1-\delta \right)}^{\frac{u}{k}}-{\delta }^{\frac{u}{k}}\right)\left(\delta \right|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right){|}^q+\left(1-\delta \right)|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right){|}^q)d\delta \hfill \\ \hfill & +{\int }_{\frac{1}{2}}^1\left({\delta }^{\frac{u}{k}}-{\left(1-\delta \right)}^{\frac{u}{k}}\right)\left(\delta \right|{\left(\xi \circ {\varrho }^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right){|}^q+\left(1-\delta \right)|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right){|}^q)d\delta .\hfill \end{array}$$

(9)

This can also be written as

$$\begin{array}{cc}\hfill & =|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right){|}^q\left({\int }_o^{\frac{1}{2}}(\delta {\left(1-\delta \right)}^{\frac{u}{k}}-{\delta }^{\frac{u}{k}+1}\right)d\delta +{\int }_{\frac{1}{2}}^1\delta \left({\delta }^{\frac{u}{k}}-{\left(1-\delta \right)}^{\frac{u}{k}}\right)d\delta )\hfill \\ \hfill & +|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right){|}^q\left({\int }_o^{\frac{1}{2}}({\left(1-\delta \right)}^{\frac{u}{k}+1}-{\delta }^{\frac{u}{k}}\left(1-\delta \right)\right)d\delta +\hfill \\ \hfill & {\int }_{\frac{1}{2}}^1\left({\delta }^{\frac{u}{k}}\left(1-\delta \right)-{\left(1-\delta \right)}^{\frac{u}{k}+1}\right)d\delta )\hfill \\ \hfill & =|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right){|}^q\frac{1}{(\frac{u}{k}+1)}\left(1-\frac{1}{2^{\frac{u}{k}}}\right)+\hfill \\ \hfill & |{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right){|}^q\frac{1}{(\frac{u}{k}+1)}\left(1-\frac{1}{2^{\frac{u}{k}}}\right).\hfill \end{array}$$

(10)

Using (8)–(10) in (7) and some computations gives us our desired result. □

In the next examples, we discuss the applications of Theorem 4 for some specific functions.

Example 9.

By substituting ${\varrho }^{s+1}\left(x\right)=e^x$ in (6), we obtain

$$\begin{array}{cc}\hfill & |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{{\left(s+1\right)}^{\frac{u}{k}}{\Gamma }_k\left(u+k\right)}{2{\left(e^n-e^m\right)}^{\frac{u}{k}}}\left({}_k^s{I}_{{e^m}^{+}}^u\xi \left(n\right)+{}_k^s{I}_{{e^n}^{-}}^u\xi \left(m\right)\right)|\hfill \\ \hfill & \le \frac{|e^n-e^m|}{2^{\frac{1}{q}}(\frac{u}{k}+1)}\left(1-\frac{1}{2^{\frac{u}{k}}}\right){\left(|e^{-m}{\xi }^{\prime }\left(m\right){|}^q+|e^{-n}{\xi }^{\prime }\left(n\right){|}^q\right)}^{\frac{1}{q}}.\hfill \end{array}$$

Example 10.

By substituting ${\varrho }^{s+1}\left(x\right)=x$ in (6), we obtain

$$\begin{array}{cc}\hfill & |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{{\left(s+1\right)}^{\frac{u}{k}}{\Gamma }_k\left(u+k\right)}{2{\left(m-n\right)}^{\frac{u}{k}}}\left({}_k^s{I}_{m^{+}}^u\xi \left(n\right){+}_k^s{I}_{n^{-}}^u\xi \left(m\right)\right)|\hfill \\ \hfill & \le \frac{|m-n|}{2^{\frac{1}{q}}(\frac{u}{k}+1)}\left(1-\frac{1}{2^{\frac{u}{k}}}\right){\left(|{\xi }^{\prime }\left(m\right){|}^q+|{\xi }^{\prime }\left(n\right){|}^q\right)}^{\frac{1}{q}}.\hfill \end{array}$$

Example 11.

By substituting ${\varrho }^{s+1}\left(x\right)={\displaystyle \frac{1}{x}}$ in (6), we obtain

$$\begin{array}{cc}\hfill & |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{{\left(s+1\right)}^{\frac{u}{k}}{\Gamma }_k\left(u+k\right)}{2}{\left(\frac{mn}{n-m}\right)}^{\frac{u}{k}}\left({}_k^s{I}_{{\frac{1}{m}}^{-}}^{\frac{u}{k}}\xi \circ g\left(\frac{1}{n}\right){+}_k^s{I}_{{\frac{1}{n}}^{-}}^{\frac{u}{k}}\xi \circ g(\frac{1}{m}\right))\hfill \\ \hfill & \le \frac{|m-n|}{2^{\frac{1}{q}}\left|mn\right|(\frac{u}{k}+1)}\left(1-\frac{1}{2^{\frac{u}{k}}}\right){\left(m^{2q}|{\xi }^{\prime }{\varrho }^{s+1}\left(m\right){|}^q+n^{2q}|{\xi }^{\prime }{\varrho }^{s+1}\left(n\right){|}^q\right)}^{\frac{1}{q}}.\hfill \end{array}$$

Example 12.

By substituting ${\varrho }^{s+1}\left(x\right)=ln\left(x\right)$ in (6), we obtain

$$\begin{array}{cc}\hfill & |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{{\left(s+1\right)}^{\frac{u}{k}}{\Gamma }_k\left(u+k\right)}{2{\left(lnn-lnm\right)}^{\frac{u}{k}}}\left({}_k^s{I}_{{lnm}^{+}}^u\xi \left(n\right)+{}_k^s{I}_{{lnn}^{-}}^u\xi \left(m\right)\right)|\hfill \\ \hfill & \le \frac{|ln(n)-ln(m\left)\right|}{2^{\frac{1}{q}}(\frac{u}{k}+1)}\left(1-\frac{1}{2^{\frac{u}{k}}}\right){\left(m^q|{\xi }^{\prime }{\varrho }^{s+1}\left(m\right){|}^q+n^q|{\xi }^{\prime }{\varrho }^{s+1}\left(n\right){|}^q\right)}^{\frac{1}{q}}.\hfill \end{array}$$

Example 13.

By substituting ${\varrho }^{s+1}\left(x\right)=x^r$ where $r\ne 0$ in (6), we obtain

$$\begin{array}{cc}\hfill & |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{r^{\frac{u}{k}}{\left(s+1\right)}^{\frac{u}{k}}{\Gamma }_k\left(u+k\right)}{2{(n^r-m^r)}^{\frac{u}{k}}}\left({}_k^s{I}_{m^{+}}^u\xi \left(n\right)+{}_k^s{I}_{n^{-}}^u\xi \left(m\right)\right)|\hfill \\ \hfill & \le \frac{|n^r-m^r|}{2^{\frac{1}{q}}\left|r\right|(\frac{u}{k}+1)}\left(1-\frac{1}{2^{\frac{u}{k}}}\right)\left(m^{(1-r)q}|{\xi }^{\prime }{\varrho }^{s+1}\left(m\right){|}^q+n^{(1-r)q}|{\xi }^{\prime }{\varrho }^{s+1}\left(n\right){|}^q\right).\hfill \end{array}$$

The next theorem is proved with the help of the following lemma.

Lemma 2

([35]). For $0<\beta <1$ and $0\le m<n$, we have

$$\begin{array}{c}\hfill |m^{\beta }-n^{\beta }|\le {\left(n-m\right)}^{\beta }.\end{array}$$

(11)

Theorem 5.

Let the two real mappings ξ and ϱ be defined on $[m,n]$ with $m<n$ such that ϱ has the strictly monotone property. If $\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}$ is differentiable and ${(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1})}^{\prime }\in L[m,n]$, then the inequality

$$\begin{array}{cc}\hfill & |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{{\left(s+1\right)}^{\frac{u}{k}}{\Gamma }_k\left(u+k\right)}{2{({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right))}^{\frac{u}{k}}}\left({}_k^s{I}_{{\varrho }^{s+1}{\left(m\right)}^{+}}^u\xi \left(n\right){+}_k^s{I}_{{\varrho }^{s+1}{\left(n\right)}^{-}}^u\xi \left(m\right)\right)|\hfill \\ \hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{2^{1+\frac{1}{q}}{(\frac{up}{k}+1)}^{\frac{1}{p}}}{\left(|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right){|}^q+|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right){|}^q\right)}^{\frac{1}{q}}\hfill \end{array}$$

(12)

holds whenever $|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{|}^q$, where $q>1$ and ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$, is convex.

Proof. 

Using Hölder’s inequality and simple calculus on the right-hand side of Lemma 1 gives us

$$\begin{array}{cc}\hfill & |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{2{({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right))}^{\frac{u}{k}}}\left({}_k^s{I}_{{\varrho }^{s+1}{\left(m\right)}^{+}}^u\xi \left(n\right){+}_k^s{I}_{{\varrho }^{s+1}{\left(n\right)}^{-}}^u\xi \left(m\right)\right)|\hfill \\ \hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{2}({\int }_0^1|{(1-\delta )}^{\frac{u}{k}}-{\delta }^{\frac{u}{k}}{|}^p{)}^{\frac{1}{p}}\hfill \\ \hfill & \times \left({\int }_0^1\right|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\left(\delta {\varrho }^{s+1}\left(m\right)+(1-\delta ){\varrho }^{s+1}\left(n\right)\right){|}^{q}d\delta )}^{\frac{1}{q}},\hfill \end{array}$$

(13)

We apply Lemma 2 and obtain

$$\begin{array}{cc}\hfill & {\int }_0^1|{\left(1-\delta \right)}^{\frac{u}{k}}-{\delta }^{\frac{u}{k}}{|}^{p}d\delta \le {\int }_0^1{\left(1-2\delta \right)}^{\frac{up}{k}}d\delta \hfill \\ \hfill & {\int }_0^{\frac{1}{2}}{\left(1-2\delta \right)}^{\frac{up}{k}}d\delta +{\int }_{\frac{1}{2}}^1{\left(2\delta -1\right)}^{\frac{up}{k}}d\delta =\frac{1}{\frac{up}{k}+1}.\hfill \end{array}$$

Now, using the fact that $|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{|}^q$ is convex, we obtain

$$\begin{array}{cc}\hfill & {\int }_0^1|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left(\delta {\varrho }^{s+1}\left(m\right)+(1-\delta ){\varrho }^{s+1}\left(n\right)\right){|}^{q}d\delta \hfill \\ \hfill & \le {\int }_0^1\left(\delta \right|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right){|}^q+\left(1-\delta \right)|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right){|}^q)d\delta \hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left(|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right){|}^q+|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right){|}^q\right).\hfill \end{array}$$

Hence we obtain our desired result (12) after making the computations above. □

In the next examples, we discuss the applications of Theorem 5 for some specific functions.

Example 14.

By substituting ${\varrho }^{s+1}\left(x\right)=e^x$ in (12), we obtain

$$\begin{array}{cc}\hfill & |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{{\left(s+1\right)}^{\frac{u}{k}}{\Gamma }_k\left(u+k\right)}{{2\left(e^n-e^m\right)}^{\frac{u}{k}}}\left({}_k^s{I}_{{e^m}^{+}}^u\xi \left(n\right)+{}_k^s{I}_{{e^n}^{-}}^u\xi \left(m\right)\right)|\hfill \\ \hfill & \le \frac{|e^n-e^m|}{2^{1+\frac{1}{q}}{(\frac{up}{k}+1)}^{\frac{1}{p}}}{\left(|e^m{\xi }^{\prime }\left(m\right){|}^q+|e^n{\xi }^{\prime }\left(n\right){|}^q\right)}^{\frac{1}{q}}.\hfill \end{array}$$

Example 15.

By substituting ${\varrho }^{s+1}\left(x\right)={\displaystyle \frac{1}{x}}$ in (12), we obtain the inequality

$$\begin{array}{cc}\hfill & |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{{\left(s+1\right)}^{\frac{u}{k}}{\Gamma }_k\left(u+k\right){\left(mn\right)}^{\frac{u}{k}}}{2{\left(n-m\right)}^{\frac{u}{k}}}\left({}_k^s{I}_{{\frac{1}{m}}^{-}}^u\left(\xi \circ g\right)\left(\frac{1}{n}\right){+}_k^s{I}_{{\frac{1}{n}}^{-}}^u\left(\xi \circ g\right)\left(\frac{1}{m}\right)\right)|\hfill \\ \hfill & \le \frac{|m-n|}{2^{1+\frac{1}{q}}{(\frac{up}{k}+1)}^{\frac{1}{p}}\left|mn\right|}{\left(m^{2q}|{\xi }^{\prime }\left(m\right){|}^q+n^{2q}|{\xi }^{\prime }\left(n\right){|}^q\right)}^{\frac{1}{q}},\hfill \end{array}$$

where $g\left(t\right)=\frac{1}{t}.$

Example 16.

By substituting ${\varrho }^{s+1}\left(x\right)=x^r$ where $r\ne 0$ in (12), we obtain the following inequality

$$\begin{array}{cc}\hfill & |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{r^{\frac{u}{k}}{\left(s+1\right)}^{\frac{u}{k}}{\Gamma }_k\left(u+k\right)}{2\left(n^r-m^r\right)}\left({}_k^s{I}_{m^{+}}^u\xi \left(n\right)+{}_k^s{I}_{n^{-}}^u\xi \left(m\right)\right)|\hfill \\ \hfill & \le \frac{|n^r-m^r|}{2^{1+\frac{1}{q}}\left|r\right|{(\frac{up}{k}+1)}^{\frac{1}{p}}}{\left(m^{(1-r)q}|{\xi }^{\prime }\left(m\right){|}^q+n^{(1-r)q}|{\xi }^{\prime }\left(n\right){|}^q\right)}^{\frac{1}{q}}.\hfill \end{array}$$

Example 17.

By substituting ${\varrho }^{s+1}\left(x\right)=ln\left(x\right)$ in (12), we obtain

$$\begin{array}{cc}\hfill & |\frac{\xi \left(m\right)+\xi \left(n\right)}{2}-\frac{{\left(s+1\right)}^{\frac{u}{k}}{\Gamma }_k\left(u+k\right)}{{2\left(lnn-lnm\right)}^{\frac{u}{k}}}\left({}_k^s{I}_{ln{m}^{+}}^u\xi \left(n\right)+{}_k^s{I}_{ln{n}^{-}}^u\xi \left(m\right)\right)|\hfill \\ \hfill & \le \frac{ln\left(n\right)-ln\left(m\right)}{2^{1+\frac{1}{q}}{(\frac{up}{k}+1)}^{\frac{1}{p}}}{\left(m^q|{\xi }^{\prime }\left(m\right){|}^q+n^q|{\xi }^{\prime }\left(n\right){|}^q\right)}^{\frac{1}{q}}.\hfill \end{array}$$

The error estimation of the inequality (2) can be studied with the help of the following identity.

Lemma 3.

The equality

$$\begin{array}{cc}\hfill & \frac{2^{\frac{u}{k}-1}{\left(s+1\right)}^{\frac{u}{k}}{\Gamma }_k\left(u+k\right)}{{({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right))}^{\frac{u}{k}}}\left({}_k^s{I}_{{\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}}^{+}}^u\xi \left(n\right){+}_k^s{I}_{{\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}}^{-}}^u\xi \left(m\right)\right)\hfill \\ \hfill & -\xi ({\left({\varrho }^{s+1}\right)}^{-1}\left(\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}\right))\hfill \\ \hfill & =\frac{{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)}{4}\left({\int }_0^1{\delta }^{\frac{u}{k}}{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\frac{{\varrho }^{s+1}\left(m\right)\delta }{2}+\left(\frac{2-\delta }{2}\right){\varrho }^{s+1}\left(n\right)\right)d\delta \hfill \\ \hfill & -{\int }_0^1{\delta }^{\frac{u}{k}}{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left(\frac{{\varrho }^{s+1}\left(m\right)(2-\delta )}{2}+\left(\frac{\delta }{2}\right){\varrho }^{s+1}\left(n\right)\right)d\delta ).\hfill \end{array}$$

(14)

holds under the same assumptions as in Lemma 1.

Proof. 

Integrating by parts, we obtain

$$\begin{array}{cc}\hfill & {\int }_0^1{\delta }^{\frac{u}{k}}{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left(\frac{{\varrho }^{s+1}\left(m\right)\delta }{2}\right)+\left(\frac{2-\delta }{2}\right)\left({\varrho }^{s+1}\left(n\right)\right)d\delta \hfill \\ \hfill & =\frac{{\delta }^{\frac{u}{k}}\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)\frac{{\varrho }^{s+1}\left(m\right)\delta }{2}+\left(\frac{2-\delta }{2}\right){\varrho }^{s+1}\left(n\right)}{\frac{{\varrho }^{s+1}\left(m\right)-{\varrho }^{s+1}\left(n\right)}{2}}{|}_0^1\hfill \\ \hfill & -\frac{u}{k}{\int }_0^1{\delta }^{\frac{u}{k}-1}\frac{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)\frac{{\varrho }^{s+1}\left(m\right)\delta }{2}+\left(\frac{2-\delta }{2}\right){\varrho }^{s+1}\left(n\right)}{\frac{{\varrho }^{s+1}\left(m\right)-{\varrho }^{s+1}\left(n\right)}{2}}d\delta \hfill \\ \hfill & =-\frac{2\xi {\left({\varrho }^{s+1}\right)}^{-1}\left(\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}\right)}{({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right))}\hfill \\ \hfill & +\left(\frac{2^{\frac{u}{k}}}{{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)}\right){\int }_0^1\left({\delta }^{\frac{u}{k}}\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right){\varrho }^{s+1}\left(m\right)\frac{\delta }{2}+\left(\frac{2-\delta }{2}\right){\varrho }^{s+1}\left(n\right)\right)d\delta .\hfill \end{array}$$

(15)

The change in variable results in the following:

$$\begin{array}{c}\hfill =-\frac{2\xi \left({\left({\varrho }^{s+1}\right)}^{-1}\left(\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}\right)\right)}{\left({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)\right)}+\frac{2^{\frac{u}{k}+1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}{\left(m\right)}^{\frac{u}{k}}}\left({}_k^s{I}_{{\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}}^{+}}^u\xi \left(n\right)\right).\end{array}$$

Similarly,

$$\begin{array}{cc}\hfill & {\int }_0^1{\delta }^{\frac{u}{k}}{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left(\frac{{\varrho }^{s+1}\left(m\right)(2-\delta )}{2}+\left(\frac{\delta }{2}\right){\varrho }^{s+1}\left(n\right)\right)d\delta =\frac{2\xi \left({\left({\varrho }^{s+1}\right)}^{-1}\left(\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}\right)\right)}{({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right))}\hfill \\ \hfill & -\frac{2^{\frac{u}{k}}}{{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)}{\int }_0^1\left({\delta }^{\frac{u}{k}-1}\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)\left(\frac{{\varrho }^{s+1}\left(m\right)\delta }{2}\right)+\left(\frac{2-\delta }{2}\right){\varrho }^{s+1}\left(n\right)\right)d\delta \hfill \\ \hfill & =\frac{2\xi \left({\left({\varrho }^{s+1}\right)}^{-1}\left(\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}\right)\right)}{({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right))}+\frac{2^{\frac{u}{k}+1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}{\left(m\right)}^{\frac{u}{k}}}\left({}_k^s{I}_{{\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}}^{+}}^u\xi \left(m\right)\right).\hfill \end{array}$$

(16)

The identity (14) is established by using (15) and (16). □

The following error estimate of the HH inequality is obtained with the help of Lemma 3.

Theorem 6.

Let the assumptions of Theorem 3 be true. Then, the following inequality

$$\begin{array}{cc}\hfill & |\frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{{({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right))}^{\frac{u}{k}}}\left({}_k^s{I}_{{\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}}^{+}}^u\xi \left(n\right){+}_k^s{I}_{{\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}}^{-}}^u\xi \left(m\right)\right)\hfill \\ \hfill & -\xi ({\left({\varrho }^{s+1}\right)}^{-1}\left(\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}\right))|\hfill \\ \hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{4(\frac{u}{k}+1)}\left(\left|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right)\right|+\left|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right)\right|\right).\hfill \end{array}$$

(17)

holds.

Proof. 

By using the fact that $\left|{(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1})}^{\prime }\right|$ is convex and basic calculus, Lemma 3 implies the following:

$$\begin{array}{cc}\hfill & |\frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{{({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right))}^{\frac{u}{k}}}\left({}_k^s{I}_{{\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}}^{+}}^u\xi \left(n\right){+}_k^s{I}_{{\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}}^{-}}^u\xi \left(m\right)\right)\hfill \\ \hfill & -\xi ({\left({\varrho }^{s+1}\right)}^{-1}\left(\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}\right))|\hfill \\ \hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{4(\frac{u}{k}+1)}({\int }_0^1{|\delta }^{\frac{u}{k}}{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left(\frac{{\varrho }^{s+1}\left(m\right)\delta }{2}\right)+\left(\frac{2-\delta }{2}\right){\varrho }^{s+1}\left(n\right)|d\delta \hfill \\ \hfill & +{\int }_0^1|{\delta }^{\frac{u}{k}}{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left(\frac{{\varrho }^{s+1}\left(m\right)(2-\delta )}{2}\right)+\left(\frac{\delta }{2}\right){\varrho }^{s+1}\left(n\right)|d\delta )\hfill \\ \hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{4(\frac{u}{k}+1)}\left(\left(\right|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right)|+\left|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right)\right|){\int }_0^1{\delta }^{\frac{u}{k}}d\delta \right).\hfill \end{array}$$

Thus, a few calculations give (17). □

In the next examples, we discuss the applications of Theorem 6 for some specific functions.

Example 18.

Putting ${\varrho }^{s+1}\left(x\right)=e^x$ in (17), we obtain the inequality

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{e^n-e^m}}\right({}_k^s{I}_{{\left(\frac{e^m+e^n}{2}\right)}^{+}}^u\xi \left(n\right)+{}_k^s{I}_{{\left(\frac{e^m+e^n}{2}\right)}^{-}}^u\xi \left(m\right)-\xi \left(ln\left(\frac{e^m+e^n}{2}\right)\right)|\hfill \\ \hfill & \le \frac{|e^n-e^m|}{4(\frac{u}{k}+1)}\left(\left|e^m{\xi }^{\prime }\left(m\right)\right|+\left|e^n{\xi }^{\prime }\left(n\right)\right|\right).\hfill \end{array}$$

Example 19.

Putting ${\varrho }^{s+1}\left(x\right)=\frac{1}{x}$ in (17), we obtain the following:

$$\begin{array}{cc}\hfill & |\frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k){\left(mn\right)}^{\frac{u}{k}}}{{(n-m)}^{\frac{u}{k}}}\left({}_k^s{I}_{{\left(\frac{m+n}{2mn}\right)}^{-}}^u\xi \circ g\left(\frac{1}{n}\right){+}_k^s{I}_{{\left(\frac{m+n}{2mn}\right)}^{+}}^u\xi \circ g\left(\frac{1}{m}\right)\right)\hfill \\ \hfill & \le \frac{(n-m)}{4(\frac{u}{k}+1)\left|mn\right|}\left(m^2\left|{\xi }^{\prime }\left(m\right)\right|+n^2\left|{\xi }^{\prime }\left(n\right)\right|\right),\hfill \end{array}$$

where $g\left(\delta \right)={\displaystyle \frac{1}{\delta }}$.

Example 20.

Putting ${\varrho }^{s+1}\left(x\right)=x^r$ where $r\ne 0$ in (17), we obtain the following inequality:

$$\begin{array}{cc}\hfill & \left|\frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k){\left(r\right)}^{\frac{u}{k}}}{{(n^r-m^r)}^{\frac{u}{k}}}\right(\left({}_k^s{I}_{{\left(\frac{m^r+n^r}{2}\right)}^{{\frac{1}{r}}^{+}}}^u\xi \left(n\right){+}_k^s{I}_{{\left(\frac{m^r+n^r}{2}\right)}^{{\frac{1}{r}}^{-}}}^u\xi \left(m\right))-\xi {\left(\frac{m^r+n^r}{2}\right)}^{\frac{1}{r}}\right)|\hfill \\ \hfill & \le \frac{|n^r-m^r|(\frac{u}{k}+2)}{4(\frac{u}{k}+1)\left|r\right|}\left(m^{(1-r)}\left|{\xi }^{\prime }\left(m\right)\right|+n^{(1-r)}\left|{\xi }^{\prime }\left(n\right)\right|\right).\hfill \end{array}$$

Example 21.

Putting ${\varrho }^{s+1}\left(x\right)=ln\left(x\right)$ in (17), we obtain the inequality

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{ln\left(n\right)-ln\left(m\right)}}\right({}_k^s{I}_{{\left(\frac{ln\left(m\right)+ln\left(n\right)}{2}\right)}^{+}}^u\xi \left(n\right)+{}_k^s{I}_{{\left(\frac{ln\left(m\right)+ln\left(n\right)}{2}\right)}^{-}}^u\xi \left(m\right)-\xi \left(exp\left(\frac{ln\left(m\right)+ln\left(n\right)}{2}\right)\right)|\hfill \\ \hfill & \le \frac{|ln(n)-ln(m\left)\right|}{4(\frac{u}{k}+1)}\left(m\left|{\xi }^{\prime }\left(m\right)\right|+n\left|{\xi }^{\prime }\left(n\right)\right|\right).\hfill \end{array}$$

Next, we give another theorem.

Theorem 7.

Let the assumptions of Theorem 4 be true. Then, the following inequality

$$\begin{array}{cc}\hfill & |\frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{{\left({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)\right)}^{\frac{u}{k}}}\left({}_k^s{I}_{{\left(\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}\right)}^{+}}^u\xi \left(n\right)+{}_k^s{I}_{{\left(\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}\right)}^{-}}^u\xi \left(m\right)\right)\hfill \\ \hfill & -\xi ({\varrho }^{s+1}{)}^{-1}\left(\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}\right)|\hfill \\ \hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{2^{2+\frac{1}{q}}(\frac{u}{k}+1){(\frac{u}{k}+2)}^{\frac{1}{q}}}\left[\right(|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right){|}^q\left(\frac{u}{k}+1\right)\hfill \\ \hfill & +|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right){|}^q\left(\frac{u}{k}+3\right){)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right){|}^q\left(\frac{u}{k}+3\right)+|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right){|}^q\left(\frac{u}{k}+1\right)\right)}^{\frac{1}{q}}].\hfill \end{array}$$

(18)

holds.

Proof. 

If $q=1$, then the above inequality can be obtained using basic calculus in Lemma 3 and $|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{|}^q$ is convex.

Now, suppose that $q>1$. The right-hand side of Lemma 3, along with the power mean inequality and simple calculus, imply the following inequality:

$$\begin{array}{cc}\hfill & |\frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{{\left({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)\right)}^{\frac{u}{k}}}\left({}_k^s{I}_{{\left(\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}\right)}^{+}}^u\xi \left(n\right){+}_k^s{I}_{{\left(\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}\right)}^{-}}^u\xi \left(m\right)\right)\hfill \\ \hfill & -\xi ({\left({\varrho }^{s+1}\right)}^{-1}\left(\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}\right))|\hfill \\ \hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{4}{\left({\int }_0^1{\delta }^{\frac{u}{k}}d\delta \right)}^{1-\frac{1}{q}}\hfill \\ \hfill & \times \left({\int }_0^1\right(|{\delta }^{\frac{u}{k}}{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left(\frac{{\varrho }^{s+1}\left(m\right)\delta }{2}+\left(\frac{2-\delta }{2}\right){\varrho }^{s+1}\left(n\right)\right){|}^{q}d\delta \hfill \\ \hfill & +{\int }_0^1\left|{\delta }^{\frac{u}{k}}{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\right(\frac{{\varrho }^{s+1}\left(m\right)\left(2-\delta \right)}{2}+\left(\frac{\delta }{2}\right){\varrho }^{s+1}\left(n\right)){|}^{q}d\delta {)}^{\frac{1}{q}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{4{(\frac{u}{k}+1)}^{1-\frac{1}{q}}}\hfill \\ \hfill & \times [{\left(|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right){|}^q{\int }_0^1\frac{{\delta }^{\frac{u}{k}+1}}{2}d\delta +|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right){|}^q{\int }_0^1\frac{\left(2-\delta \right){\delta }^{\frac{u}{k}}}{2}d\delta \right)}^{1-\frac{1}{q}}\hfill \\ \hfill & +{\left(|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right){|}^q{\int }_0^1\frac{\left(2-\delta \right){\delta }^{\frac{u}{k}}}{2}d\delta +|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right){|}^q{\int }_0^1\frac{{\delta }^{\frac{u}{k}+1}}{2}d\delta \right)}^{1-\frac{1}{q}}]\hfill \\ \hfill & =\frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{2^{2+\frac{1}{q}}(\frac{u}{k}+1){(\frac{u}{k}+2)}^{\frac{1}{q}}}\left[\right(|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right){|}^q\left(\frac{u}{k}+1\right)+\hfill \\ \hfill & |{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right){|}^q\left(\frac{u}{k}+3\right){)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right){|}^q\left(\frac{u}{k}+3\right)+|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right){|}^q\left(\frac{u}{k}+1\right)\right)}^{\frac{1}{q}}].\hfill \end{array}$$

□

In the next examples, we discuss the applications of Theorem 7 for some specific functions.

Example 22.

By substituting ${\varrho }^{s+1}\left(x\right)=e^x$ in (18), we obtain

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{e^n-e^m}}\right({}_k^s{I}_{{\left(\frac{e^m+e^n}{2}\right)}^{+}}^u\xi \left(n\right)+{}_k^s{I}_{{\left(\frac{e^m+e^n}{2}\right)}^{-}}^u\xi \left(m\right)-\xi \left(ln\left({\displaystyle \frac{e^m+e^n}{2}}\right)\right)|\hfill \\ \hfill & \le {\displaystyle \frac{|e^n-e^m|}{2^{2+\frac{1}{q}}(\frac{u}{k}+1){(\frac{u}{k}+2)}^{\frac{1}{q}}}}\hfill \\ \hfill & \times [{\left(|e^m{\xi }^{\prime }\left(m\right){|}^q\left({\displaystyle \frac{u}{k}}+1\right)+|e^n{\xi }^{\prime }\left(n\right){|}^q\left({\displaystyle \frac{u}{k}}+3\right)\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(|e^m{\xi }^{\prime }\left(m\right){|}^q\left({\displaystyle \frac{u}{k}}+3\right)+|e^n{\xi }^{\prime }\left(n\right){|}^q\left(\frac{u}{k}+1\right)\right)}^{\frac{1}{q}}].\hfill \end{array}$$

Example 23.

By substituting ${\varrho }^{s+1}\left(x\right)={\displaystyle \frac{1}{x}}$ in (18), we obtain

$$\begin{array}{cc}\hfill & \left|\frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k){\left(mn\right)}^{\frac{u}{k}}}{{(n-m)}^{\frac{u}{k}}}\left({}_k^s{I}_{{\left(\frac{m+n}{2mn}\right)}^{-}}^u\xi \circ g\left(\frac{1}{n}\right){+}_k^s{I}_{{\left(\frac{m+n}{2mn}\right)}^{+}}^u\xi \circ g\left(\frac{1}{m}\right)\right)\right|\hfill \\ \hfill & \le \frac{|m-n|}{2^{2+\frac{1}{q}}(\frac{u}{k}+1){(\frac{u}{k}+2)}^{\frac{1}{q}}\left|mn\right|}\hfill \\ \hfill & \times [{\left(m^{2q}|{\xi }^{\prime }\left(m\right){|}^q\left({\displaystyle \frac{u}{k}}+1\right)+n^{2q}|{\xi }^{\prime }\left(n\right){|}^q\left(\frac{u}{k}+3\right)\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(m^{2q}|{\xi }^{\prime }\left(m\right){|}^q\left(\frac{u}{k}+3\right)+n^{2q}|{\xi }^{\prime }\left(n\right){|}^q\left(\frac{u}{k}+1\right)\right)}^{\frac{1}{q}}],\hfill \end{array}$$

where $g\left(\nu \right)=\frac{1}{\nu }$.

Example 24.

By substituting ${\varrho }^{s+1}\left(x\right)=x^r$ where $r\ne 0$ in (18), we obtain the inequality

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k){\left(r\right)}^{\frac{u}{k}}}{{(n^r-m^r)}^{\frac{u}{k}}}}\left({}_k^s{I}_{{\left(\frac{m^r+n^r}{2}\right)}^{\frac{1}{r}+}}^u\xi \left(n\right){+}_k^s{I}_{{\left(\frac{m^r+n^r}{2}\right)}^{\frac{1}{r}-}}^u\xi \left(m\right)\right)-\xi {\left(\frac{m^r+n^r}{2}\right)}^{\frac{1}{r}}|\hfill \\ \hfill & \le \frac{|n^r-m^r|}{2^{2+\frac{1}{q}}(\frac{u}{k}+1){(\frac{u}{k}+2)}^{\frac{1}{q}}\left|r\right|}\hfill \\ \hfill & \times [{\left(m^{q(1-r)}|{\xi }^{\prime }\left(m\right){|}^q\left(\frac{u}{k}+1\right)+n^{q(1-r)}|{\xi }^{\prime }\left(n\right){|}^q\left(\frac{u}{k}+3\right)\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(m^{q(1-r)}|{\xi }^{\prime }\left(m\right){|}^q\left(\frac{u}{k}+3\right)+n^{q(1-r)}|{\xi }^{\prime }\left(n\right){|}^q\left(\frac{u}{k}+1\right)\right)}^{\frac{1}{q}}].\hfill \end{array}$$

Example 25.

By substituting ${\varrho }^{s+1}\left(x\right)=ln\left(x\right)$ in (18), we obtain

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{ln\left(n\right)-ln\left(m\right)}}\right({}_k^s{I}_{{\left(\frac{ln\left(m\right)+ln\left(n\right)}{2}\right)}^{+}}^u\xi \left(n\right)+{}_k^s{I}_{{\left(\frac{ln\left(m\right)+ln\left(n\right)}{2}\right)}^{-}}^u\xi \left(m\right)-\xi \left(exp\left({\displaystyle \frac{ln\left(m\right)+ln\left(n\right)}{2}}\right)\right)|\hfill \\ \hfill & \le {\displaystyle \frac{|ln(n)-ln(m\left)\right|}{2^{2+\frac{1}{q}}(\frac{u}{k}+1){(\frac{u}{k}+2)}^{\frac{1}{q}}}}\hfill \\ \hfill & \times [{\left(m^q|{\xi }^{\prime }\left(m\right){|}^q\left({\displaystyle \frac{u}{k}}+1\right)+n^q|{\xi }^{\prime }\left(n\right){|}^q\left({\displaystyle \frac{u}{k}}+3\right)\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(m^q|{\xi }^{\prime }\left(m\right){|}^q\left({\displaystyle \frac{u}{k}}+3\right)+n^q|{\xi }^{\prime }\left(n\right){|}^q\left(\frac{u}{k}+1\right)\right)}^{\frac{1}{q}}].\hfill \end{array}$$

Theorem 8.

The following inequality

$$\begin{array}{cc}\hfill & |\frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{{\left({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)\right)}^{\frac{u}{k}}}\left({}_k^s{I}_{{\left(\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}\right)}^{+}}^u\xi \left(n\right){+}_k^s{I}_{{\left(\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}\right)}^{-}}^u\xi \left(m\right)\right)\hfill \\ \hfill & -\xi ({\left({\varrho }^{s+1}\right)}^{-1}\left(\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}\right))|\hfill \\ \hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{4^{1-\frac{1}{p}}{(\frac{up}{k}+1)}^{\frac{1}{p}}}\left(\left|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right)\right|+\left|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right)\right|\right).\hfill \end{array}$$

(19)

holds under the same assumptions as in Theorem 5.

Proof. 

By using Hölder’s inequality and basic calculus in Lemma 3, we obtain

$$\begin{array}{cc}\hfill & |\frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{{\left({\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)\right)}^{\frac{u}{k}}}\left({}_k^s{I}_{{\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}}^{+}}^u\xi \left(n\right){+}_k^s{I}_{{\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}}^{-}}^u\xi \left(m\right)\right)\hfill \\ \hfill & -\xi ({\left({\varrho }^{s+1}\right)}^{-1}\left(\frac{{\varrho }^{s+1}\left(m\right)+{\varrho }^{s+1}\left(n\right)}{2}\right))|\hfill \\ \hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{4}{\left({\int }_0^1{\delta }^{\frac{up}{k}}d\delta \right)}^{\frac{1}{p}}\hfill \\ \hfill & \times \left[{\int }_0^1\right|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left(\frac{{\varrho }^{s+1}\left(m\right)\delta }{2}\right)+\left(\frac{2-\delta }{2}\right){\varrho }^{s+1}\left(n\right)|d\delta \hfill \\ \hfill & +{\int }_0^1|{\delta }^{\frac{u}{k}}{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }\left(\frac{{\varrho }^{s+1}\left(m\right)\left(2-\delta \right)}{2}\right)+\left(\frac{\delta }{2}\right){\varrho }^{s+1}\left(n\right){|}^{q}d\delta ].\hfill \end{array}$$

(20)

As $|{(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1})}^{\prime }|$ is convex, the right-hand side of the inequality (20) takes the following form:

$$\begin{array}{cc}\hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{4{(\frac{up}{k}+1)}^{\frac{1}{p}}}(|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right){|}^q{\int }_0^1\frac{\delta }{2}d\delta +\hfill \\ \hfill & |{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right){|}^q{\int }_0^1\left(\frac{2-\delta }{2}\right)d\delta {)}^{\frac{1}{q}}\hfill \\ \hfill & +|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right){|}^q{\int }_0^1\left(\frac{2-\delta }{2}\right)d\delta +\left(|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right)\right){|}^q{\int }_0^1\frac{\delta }{2}d\delta {)}^{\frac{1}{q}}\hfill \\ \hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{4{(\frac{up}{k}+1)}^{\frac{1}{p}}}[{\left(\frac{|{(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1})}^{\prime }{\varrho }^{s+1}{\left(m\right)|}^q+3|{(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1})}^{\prime }{\varrho }^{s+1}{\left(n\right)|}^q}{4}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{3|{(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1})}^{\prime }{\varrho }^{s+1}{\left(m\right)|}^q+|{(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1})}^{\prime }{\varrho }^{s+1}{\left(n\right)|}^q}{4}\right)}^{\frac{1}{q}}]\hfill \\ \hfill & \le \frac{|{\varrho }^{s+1}\left(n\right)-{\varrho }^{s+1}\left(m\right)|}{4^{1-\frac{1}{p}}{(\frac{up}{k}+1)}^{\frac{1}{p}}}\left(\left|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(m\right)\right|+\left|{\left(\xi \circ {\left({\varrho }^{s+1}\right)}^{-1}\right)}^{\prime }{\varrho }^{s+1}\left(n\right)\right|\right).\hfill \end{array}$$

In the above computations, we use $m^p+n^q\le {(m+n)}^q$ where $q>1$ and $m,n\ge 0$ to obtain our desired result. □

In the next examples, we discuss the applications of Theorem 8 for some specific functions.

Example 26.

Putting ${\varrho }^{s+1}\left(x\right)=e^x$ in (19), we obtain

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{e^n-e^m}}\right({}_k^s{I}_{{\left(\frac{e^m+e^n}{2}\right)}^{+}}^u\xi \left(n\right)+{}_k^s{I}_{{\left(\frac{e^m+e^n}{2}\right)}^{-}}^u\xi \left(m\right)-\xi \left(ln\left(\frac{e^m+e^n}{2}\right)\right)|\hfill \\ \hfill & \le \frac{|e^n-e^m|}{4^{1-\frac{1}{p}}{(\frac{up}{k}+1)}^{\frac{1}{p}}}\left(e^m\left|{\xi }^{\prime }\left(m\right)\right|+e^n\left|{\xi }^{\prime }\left(n\right)\right|\right).\hfill \end{array}$$

Example 27.

By substituting ${\varrho }^{s+1}\left(x\right)={\displaystyle \frac{1}{x}}$ in (19), we obtain the inequality

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k){\left(mn\right)}^{\frac{u}{k}}}{{(n-m)}^{\frac{u}{k}}}}\left({}_k^s{I}_{{\left(\frac{m+n}{2mn}\right)}^{-}}^u\xi \circ g\left(\frac{1}{n}\right){+}_k^s{I}_{{\left(\frac{m+n}{2mn}\right)}^{+}}^u\xi \circ g\left(\frac{1}{m}\right)\right)\hfill \\ \hfill & \le \frac{|m-n|}{4^{1-\frac{1}{p}}{(\frac{up}{k}+1)}^{\frac{1}{p}}\left|mn\right|}\left(m^2\left|{\xi }^{\prime }\left(m\right)\right|+n^2\left|{\xi }^{\prime }\left(n\right)\right|\right),\hfill \end{array}$$

where $g\left(\delta \right)={\displaystyle \frac{1}{\delta }}$.

Example 28.

By using ${\varrho }^{s+1}\left(x\right)=x^r$ in (19), we obtain

$$\begin{array}{cc}\hfill & \left|\frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k){\left(r\right)}^{\frac{u}{k}}}{{(n^r-m^r)}^{\frac{u}{k}}}\left(\left({}_k^s{I}_{{\left(\frac{m^r+n^r}{2}\right)}^{\frac{1}{r}+}}^u\xi \left(n\right){+}_k^s{I}_{({\frac{m^r+n^r}{2})}^{\frac{1}{r}-}}^u\xi \left(m\right)\right)-\xi {\left(\frac{m^r+n^r}{2}\right)}^{\frac{1}{r}}\right)\right|\hfill \\ \hfill & \le \frac{|n^r-m^r|}{4^{1-\frac{1}{p}}\left|r\right|{(\frac{up}{k}+1)}^{\frac{1}{p}}}\left(m^{1-r}\left|{\xi }^{\prime }\left(m\right)\right|+n^{1-r}\left|{\xi }^{\prime }\left(n\right)\right|\right).\hfill \end{array}$$

Example 29.

Putting ${\varrho }^{s+1}\left(x\right)=ln\left(x\right)$ in (19), we obtain

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{2^{\frac{u}{k}-1}{(s+1)}^{\frac{u}{k}}{\Gamma }_k(u+k)}{ln\left(n\right)-ln\left(m\right)}}\right({}_k^s{I}_{{\left(\frac{ln\left(m\right)+ln\left(n\right)}{2}\right)}^{+}}^u\xi \left(n\right)+{}_k^s{I}_{{\left(\frac{ln\left(m\right)+ln\left(n\right)}{2}\right)}^{-}}^u\xi \left(m\right)-\xi \left(exp\left(\frac{ln\left(m\right)+ln\left(n\right)}{2}\right)\right)|\hfill \\ \hfill & \le \frac{|ln(n)-ln(m\left)\right|}{4^{1-\frac{1}{p}}{(\frac{up}{k}+1)}^{\frac{1}{p}}}\left(m\left|{\xi }^{\prime }\left(m\right)\right|+n\left|{\xi }^{\prime }\left(n\right)\right|\right).\hfill \end{array}$$

4. Concluding Remarks

Several generalized novel mean-type inequalities were obtained via certain convex functions using fractional calculus and convexity theory. Using the generalized form of the RL-fractional integral, namely the $(k,s)$-RL-fractional integral, we established various mean-type inequalities for strictly monotone functions in this article. The main results were formulated after first establishing a special identity. Through different choices of strictly monotone functions, the corresponding inequalities were also produced. The findings of this article can be reduced to that of [22] by taking $k=1$ and $s=0$. The obtained results are hopefully helpful in the field of modified scientific disciplines.