Midpoint Inequalities in Fractional Calculus Defined Using Positive Weighted Symmetry Function Kernels

Abstract

The aim of our study is to establish, for convex functions on an interval, a midpoint version of the fractional HHF type inequality. The corresponding fractional integral has a symmetric weight function composed with an increasing function as integral kernel. We also consider a midpoint identity and establish some related inequalities based on this identity. Some special cases can be considered from our main results. These results confirm the generality of our attempt.

1. Introduction

Let $\mathcal{J}\subset \mathcal{R}$ be an interval and let $\mathrm{u}:\mathcal{J}\to \mathcal{R}$ be a continuous function. Then, the function $\mathrm{u}$ is called convex if it satisfies

$$\begin{array}{c}\hfill \mathrm{u}(\kappa {\mathbf{c}}_1+(1-\kappa ){\mathbf{c}}_2)\le \kappa \mathrm{u}\left({\mathbf{c}}_1\right)+(1-\kappa )\mathrm{u}\left({\mathbf{c}}_2\right),\forall {\mathbf{c}}_1,{\mathbf{c}}_2\in \mathcal{J}\mathrm{and}\kappa \in [0,1].\end{array}$$

(1)

The function $\mathrm{u}$ is called concave whenever $-\mathrm{u}$ is convex.

For convex functions $\mathrm{u}:\mathcal{J}\to \mathcal{R}$, there is an important integral inequality in the literature, namely the Hermite–Hadamard or, briefly, the $\mathtt{HH}$ integral inequality, which is given by [1]:

$$\begin{array}{cc}\hfill \mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)& \le \frac{1}{{\mathbf{c}}_2-{\mathbf{c}}_1}{\int }_{{\mathbf{c}}_1}^{{\mathbf{c}}_2}\mathrm{u}\left(x\right)dx\le \frac{\mathrm{u}\left({\mathbf{c}}_1\right)+\mathrm{u}\left({\mathbf{c}}_2\right)}{2},\hfill \end{array}$$

(2)

where ${\mathbf{c}}_1<{\mathbf{c}}_2$ belong to $\mathcal{J}.$ In the literature, one can observe that the $\mathtt{HH}$ integral inequality (2) has been applied to different classes of convexity such as $GA$–convexity [2], quasi-convexity [3,4], s–convexity [5], $(\alpha ,m)$–convexity [6], exponentially convexity [7,8], $MT$–convexity [9], and the readers can consult [10,11] to find other types.

As we know, fractional calculus is a generalized form of integer order calculus. Various forms of fractional derivatives including $\mathtt{RL}$, Hadamard, Caputo, Caputo–Hadamard, Riesz, $\psi$–$\mathtt{RL}$, Prabhakar, and weighted versions [12,13,14,15,16] have been developed to date. Most of these versions are described in the $\mathtt{RL}$ sense based on the corresponding fractional integral. Many integer-order integral inequalities such as Ostrowski [17], Simpson [18], Hardy [19], Olsen [20], Gagliardo–Nirenberg [21], Opial [22,23] and Rozanova [24] have been generalized and reformulated from the fractional point of view.

In addition, in 2013, the $\mathtt{HH}$ integral inequality (2) was generalized and reformulated by Sarikaya et al. [25] in terms of $\mathtt{RL}$ fractional integrals. Their result is given by:

$$\begin{array}{cc}\hfill \mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)& \le \frac{\mathsf{\Gamma }(\nu +1)}{2{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu }}\left[{{}^{RL}\mathcal{I}}_{{\mathbf{c}}_1+}^{\nu }\mathrm{u}\left({\mathbf{c}}_2\right)+{{}^{RL}\mathcal{I}}_{{\mathbf{c}}_2-}^{\nu }\mathrm{u}\left({\mathbf{c}}_1\right)\right]\le \frac{\mathrm{u}\left({\mathbf{c}}_1\right)+\mathrm{u}\left({\mathbf{c}}_2\right)}{2},\hfill \end{array}$$

(3)

where $\mathrm{u}:\mathcal{J}\to \mathcal{R}$ is assumed to be a positive convex function, continuous on the closed interval $[{\mathbf{c}}_1,{\mathbf{c}}_2],$ and for Lebesgue, almost all $x\in [{\mathbf{c}}_1,{\mathbf{c}}_2]$ when $\mathrm{u}\left(x\right)\in L^1[{\mathbf{c}}_1,{\mathbf{c}}_2]$ with ${\mathbf{c}}_1<{\mathbf{c}}_2$, where ${}^{RL}{\mathcal{I}}_{{\mathbf{c}}_1+}^{\nu }$ and ${}^{RL}{\mathcal{I}}_{{\mathbf{c}}_2-}^{\nu }$ are the left- and right-sided $\mathtt{RL}$ fractional integrals of order $\nu >0$, defined by [12]:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {{}^{RL}\mathcal{I}}_{{\mathbf{c}}_1+}^{\nu }\mathrm{u}\left(x\right)=\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\mathbf{c}}_1}^x{(x-\kappa )}^{\nu -1}\mathrm{u}\left(\kappa \right)d\kappa ,x>{\mathbf{c}}_1;\\ \hfill {{}^{RL}\mathcal{I}}_{{\mathbf{c}}_2-}^{\nu }\mathrm{u}\left(x\right)=\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_x^{{\mathbf{c}}_2}{(\kappa -x)}^{\nu -1}\mathrm{u}\left(\kappa \right)d\kappa ,x<{\mathbf{c}}_2,\end{array}\end{array}$$

(4)

respectively.

The inequality (3) is also known as the endpoint $\mathtt{HH}$ inequality due to using the ends ${\mathbf{c}}_1$, ${\mathbf{c}}_2$ of the interval.

On the other hand, the endpoint $\mathtt{HH}$ inequality (3) has been applied for various classes of convexity such as ${\lambda }_{\psi }$–convexity [26], F–convexity [27], $(\alpha ,m)$–convexity [28], $MT$–convexity [29]. The reader can find other types of convexity in the literature, which in particular, is true for [30]. In the mean time, applying the end-point $\mathtt{HH}$ inequality to other models of fractional calculus has received a huge amount of attention. For example, this is true for $\mathtt{RL}$ fractional models [31], conformable fractional models [32,33], generalized fractional models [34], $\psi$ $\mathtt{RL}$ fractional models [35,36], tempered fractional models [37], and $AB$- and Prabhakar fractional models [38].

After extending the important field of the integral inequalities in (2) and (3), a new version of the endpoint $\mathtt{HH}$ inequality (3) was found by Sarikaya and Yildirim [39], namely the midpoint $\mathtt{HH}$ inequality due to using the midpoint $\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}$ of the interval, which is given by

$$\begin{array}{c}\hfill \mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)\le \frac{2^{\nu -1}\mathsf{\Gamma }(\nu +1)}{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu }}\left[{{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}^{\nu }\mathrm{u}\left({\mathbf{c}}_2\right)+{{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu }\mathrm{u}\left({\mathbf{c}}_1\right)\right]\le \frac{\mathrm{u}\left({\mathbf{c}}_1\right)+\mathrm{u}\left({\mathbf{c}}_2\right)}{2},\end{array}$$

(5)

where the function $\mathrm{u}:[{\mathbf{c}}_1,{\mathbf{c}}_2]\to \mathcal{R}$ is convex and continuous.

Definition 1

([40]). Let $g:[{\mathbf{c}}_1,{\mathbf{c}}_2]\to [0,\infty )$ be a function. Then, we say g is symmetric with respect to $({\mathbf{c}}_1+{\mathbf{c}}_2)/2$ if

$$g({\mathbf{c}}_1+{\mathbf{c}}_2-x)=g\left(x\right),\forall x\in [{\mathbf{c}}_1,{\mathbf{c}}_2]$$

(6)

Based on above definition, in [41], Fejér found a new extension of the $\mathtt{HH}$ type inequality (2), namely the $\mathtt{HHF}$ type inequality, and the result is as follows:

$$\begin{array}{cc}\hfill \mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right){\int }_{{\mathbf{c}}_1}^{{\mathbf{c}}_2}g\left(x\right)dx& \le {\int }_{{\mathbf{c}}_1}^{{\mathbf{c}}_2}\mathrm{u}\left(x\right)g\left(x\right)dx\le \frac{\mathrm{u}\left({\mathbf{c}}_1\right)+\mathrm{u}\left({\mathbf{c}}_2\right)}{2}{\int }_{{\mathbf{c}}_1}^{{\mathbf{c}}_2}g\left(x\right)dx,\hfill \end{array}$$

(7)

where g is the integrable function, and Işcan [42] found the endpoint version of (7) in the sense of $\mathtt{RL}$ fractional integrals, which is also the extension of (3). The result is as follows:

$$\begin{array}{c}\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)\left[{{}^{RL}\mathcal{I}}_{{\mathbf{c}}_1+}^{\nu }g\left({\mathbf{c}}_2\right)+{{}^{RL}\mathcal{I}}_{{\mathbf{c}}_2-}^{\nu }g\left({\mathbf{c}}_1\right)\right]\le \left[{{}^{RL}\mathcal{I}}_{{\mathbf{c}}_1+}^{\nu }\left(\mathrm{u}g\right)\left({\mathbf{c}}_2\right)+{{}^{RL}\mathcal{I}}_{{\mathbf{c}}_2-}^{\nu }\left(\mathrm{u}g\right)\left({\mathbf{c}}_1\right)\right]\hfill \\ \hfill \le \frac{\mathrm{u}\left({\mathbf{c}}_1\right)+\mathrm{u}\left({\mathbf{c}}_2\right)}{2}\left[{{}^{RL}\mathcal{I}}_{{\mathbf{c}}_1+}^{\nu }g\left({\mathbf{c}}_2\right)+{{}^{RL}\mathcal{I}}_{{\mathbf{c}}_2-}^{\nu }g\left({\mathbf{c}}_1\right)\right],\end{array}$$

(8)

where $\mathrm{u}$ is convex and continuous and the function g belongs to $L^1[{\mathbf{c}}_1,{\mathbf{c}}_2]$ and is symmetric (see Definition 1).

It is worth mentioning that the midpoint version of (8) has not been found yet, even though many related inequalities of midpoint type were obtained in [43].

Recently, Mohammed et al. [44] found a new endpoint $\mathtt{HHF}$-inequality in terms of weighted fractional integrals with positive weighted symmetric function in a kernel, and their result is as follows:

$$\begin{array}{c}\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)\left[\left({{}_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)+}\mathcal{I}}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)+\left({\mathcal{I}}_{{\varrho }^{-1}\left({\mathbf{c}}_2\right)-}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)\right]\hfill \\ \le w\left({\mathbf{c}}_2\right)\left({{}_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)+}\mathcal{I}}_{w\circ \varrho }^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)+w\left({\mathbf{c}}_1\right)\left({{}_{w\circ \varrho }\mathcal{I}}_{{\varrho }^{-1}\left({\mathbf{c}}_2\right)-}^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)\\ \le \frac{\mathrm{u}\left({\mathbf{c}}_1\right)+\mathrm{u}\left({\mathbf{c}}_2\right)}{2}[\left({{}_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)+}\mathcal{I}}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\\ \hfill +\left({\mathcal{I}}_{{\varrho }^{-1}\left({\mathbf{c}}_2\right)-}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)].\end{array}$$

(9)

Here, $\mathrm{u}$ is a convex and continuous function, $\varrho \left(x\right)$ a monotone increasing function from the interval $({\mathbf{c}}_1,{\mathbf{c}}_2]$ onto itself with a continuous derivative ${\varrho }^{\prime }\left(x\right)$ on the open interval $({\mathbf{c}}_1,{\mathbf{c}}_2),$ and $w:[{\mathbf{c}}_1,{\mathbf{c}}_2]\to (0,\infty )$ is an integrable function, which is symmetric with respect to $({\mathbf{c}}_1+{\mathbf{c}}_2)/2,$ where ${\mathbf{c}}_1<{\mathbf{c}}_2.$

Definition 2.

Let $({\mathbf{c}}_1,{\mathbf{c}}_2)\subseteq \mathcal{R}$ and $\varrho \left(x\right)$ be an increasing positive and monotone function on the interval $({\mathbf{c}}_1,{\mathbf{c}}_2]$ with a continuous derivative ${\varrho }^{\prime }\left(x\right)$ on the open interval $({\mathbf{c}}_1,{\mathbf{c}}_2).$ Then, the left-sided and right-sided the weighted fractional integrals of a function $\mathrm{u}$ according to another function $\varrho \left(x\right)$ on $[{\mathbf{c}}_1,{\mathbf{c}}_2]$ are defined by [15]:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left({{}_{{\mathbf{c}}_1+}\mathcal{I}}_w^{\nu :\varrho }\mathrm{u}\right)\left(x\right)& =\frac{{\left[w\left(x\right)\right]}^{-1}}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\mathbf{c}}_1}^x{\varrho }^{\prime }\left(\kappa \right){(\varrho \left(x\right)-\varrho \left(\kappa \right))}^{\nu -1}\mathrm{u}\left(\kappa \right)w\left(\kappa \right)d\kappa ,\hfill \\ \hfill \left({{}_w\mathcal{I}}_{{\mathbf{c}}_2-}^{\nu :\varrho }\mathrm{u}\right)\left(x\right)& =\frac{{\left[w\left(x\right)\right]}^{-1}}{\mathsf{\Gamma }\left(\nu \right)}{\int }_x^{{\mathbf{c}}_2}{\varrho }^{\prime }\left(\kappa \right){(\varrho \left(\kappa \right)-\varrho \left(x\right))}^{\nu -1}\mathrm{u}\left(\kappa \right)w\left(\kappa \right)d\kappa ,\nu >0,\hfill \end{array}\end{array}$$

(10)

for ${\left[w\left(x\right)\right]}^{-1}\frac{1}{w\left(x\right)}$ such that $w\left(x\right)\ne 0$.

Remark 1.

From Definition 2, we can obtain the following special cases.

• If $\varrho \left(x\right)=x$ and $w\left(x\right)=1$, then the weighted fractional integrals (10) reduce to the classical $\mathtt{RL}$ fractional integrals (4).
• If $w\left(x\right)=1$, we obtain the fractional integrals of the function $\mathrm{u}$ with respect to the function $\varrho \left(x\right)$, which is defined by [13,14]:

  $$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left({{}_{{\mathbf{c}}_1+}\mathcal{I}}^{\nu :\varrho }\mathrm{u}\right)\left(x\right)& =\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\mathbf{c}}_1}^x{\varrho }^{\prime }\left(\kappa \right){(\varrho \left(x\right)-\varrho \left(\kappa \right))}^{\nu -1}\mathrm{u}\left(\kappa \right)d\kappa ,\hfill \\ \hfill \left({\mathcal{I}}_{{\mathbf{c}}_2-}^{\nu :\varrho }\mathrm{u}\right)\left(x\right)& =\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_x^{{\mathbf{c}}_2}{\varrho }^{\prime }\left(\kappa \right){(\varrho \left(\kappa \right)-\varrho \left(x\right))}^{\nu -1}\mathrm{u}\left(\kappa \right)d\kappa ,\nu >0.\hfill \end{array}\end{array}$$

  (11)

In this article, we will investigate the midpoint version of (9) and some related $\mathtt{HHF}$ inequalities by using the weighted fractional integrals (10) with positive weighted symmetric functions in the kernel.

The rest of our article is structured in the following way: In Section 2, we will prove the necessary and auxiliary lemmas, including the midpoint version of (9). In Section 3, we will prove our main results, including new midpoint fractional $\mathtt{HHF}$ integral inequalities with some related results. We will present some concluding remarks in Section 4.

2. Auxiliary Results

In this section, we prove analogues of the fractional $\mathtt{HH}$ inequalities (2)–(3) and $\mathtt{HHF}$ inequalities (7)–(8) for weighted fractional integral operators with positive weighted symmetric function kernels. Here, the main results are as follows: Theorem 1 (it is a generalisation of $\mathtt{HH}$ inequalities (2)–(3) and $\mathtt{HHF}$ inequality (7), and a reformulation of $\mathtt{HHF}$ inequality (8)) and Lemma 2 (it is a consequence of Theorem 1).

At first, we need the following lemma.

Lemma 1. 

Assume that $w:[{\mathbf{c}}_1,{\mathbf{c}}_2]\to (0,\infty )$ is an integrable function and symmetric with respect to $({\mathbf{c}}_1+{\mathbf{c}}_2)/2,{\mathbf{c}}_1<{\mathbf{c}}_2$. Then,

(i)

for each $\kappa \in [0,1]$, we have

$$w\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)=w\left(\frac{2-\kappa }{2}{\mathbf{c}}_1+\frac{\kappa }{2}{\mathbf{c}}_2\right).$$

(12)

(ii)

For $\nu >0$, we have

$$\begin{array}{c}\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)=\left({\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)\hfill \\ \hfill =\frac{1}{2}\left[\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)+\left({\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)\right].\end{array}$$

(13)

Proof. 

(i)

Let $x=\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2$. It is clear that $x\in [{\mathbf{c}}_1,{\mathbf{c}}_2]$ for each $\kappa \in [0,1]$ and that ${\mathbf{c}}_1+{\mathbf{c}}_2-x=\frac{2-\kappa }{2}{\mathbf{c}}_1+\frac{\kappa }{2}{\mathbf{c}}_2$. Then, by making use of the assumptions and Definition 1, we can obtain (12).

(ii)

The symmetry property of w leads to

$$\begin{array}{c}\hfill (w\circ \varrho )\left(\kappa \right)=w\left(\varrho \left(\kappa \right)\right)=w({\mathbf{c}}_1+{\mathbf{c}}_2-\varrho \left(\kappa \right)),\forall \kappa \in \left[{\varrho }^{-1}\left({\mathbf{c}}_1\right),{\varrho }^{-1}\left({\mathbf{c}}_2\right)\right].\end{array}$$

From above and setting $\varrho \left(x\right):={\mathbf{c}}_1+{\mathbf{c}}_2-\varrho \left(\kappa \right)$, it follows that

$$\begin{array}{cc}\hfill & \left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\hfill \\ \hfill & =\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}(w\circ \varrho )\left(x\right){\varrho }^{\prime }\left(x\right)dx\hfill \\ \hfill & =\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}{(\varrho \left(\kappa \right)-{\mathbf{c}}_1)}^{\nu -1}w({\mathbf{c}}_1+{\mathbf{c}}_2-\varrho \left(\kappa \right)){\varrho }^{\prime }\left(\kappa \right)d\kappa \hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}{(\varrho \left(\kappa \right)-{\mathbf{c}}_1)}^{\nu -1}(w\circ \varrho )\left(\kappa \right){\varrho }^{\prime }\left(\kappa \right)d\kappa \hfill \\ \hfill & =\left({\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right),\hfill \end{array}$$

which completes the desired equality (13). □

Remark 2. 

Throughout the present article, we denote ${\left[w\left(x\right)\right]}^{-1}=\frac{1}{w\left(x\right)}$ and ${\varrho }^{-1}\left(x\right)$ the inverse of the function $\varrho \left(x\right)$.

Theorem 1. 

Let $0\le {\mathbf{c}}_1<{\mathbf{c}}_2$, let $\mathrm{u}:[{\mathbf{c}}_1,{\mathbf{c}}_2]\to \mathcal{R}$ be an $L^1$ convex function and $w:[{\mathbf{c}}_1,{\mathbf{c}}_2]\to \mathcal{R}$ be an integrable, positive and weighted symmetric function with respect to $\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}$. If, in addition, ϱ is an increasing and positive function from $[{\mathbf{c}}_1,{\mathbf{c}}_2)$ onto itself such that its derivative ${\varrho }^{\prime }\left(x\right)$ is continuous on $({\mathbf{c}}_1,{\mathbf{c}}_2)$, then for $\nu >0$, the following inequalities are valid:

$$\begin{array}{c}\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)[\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\hfill \\ +\left({\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)]\le w\left({\mathbf{c}}_2\right)\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}_{w\circ \varrho }^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\\ +w\left({\mathbf{c}}_1\right)\left({{}_{w\circ \varrho }\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)\\ \le \frac{\mathrm{u}\left({\mathbf{c}}_1\right)+\mathrm{u}\left({\mathbf{c}}_2\right)}{2}[\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\\ \hfill +\left({\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)].\end{array}$$

(14)

Proof. 

The convexity of $\mathrm{u}$ on $[{\mathbf{c}}_1,{\mathbf{c}}_2]$ gives

$$\begin{array}{c}\hfill \mathrm{u}\left(\frac{x+y}{2}\right)\le \frac{\mathrm{u}\left(x\right)+\mathrm{u}\left(y\right)}{2}\mathrm{for}\mathrm{all}x,y\in [{\mathbf{c}}_1,{\mathbf{c}}_2].\end{array}$$

So, for $x=\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2$ and $y=\frac{2-\kappa }{2}{\mathbf{c}}_1+\frac{\kappa }{2}{\mathbf{c}}_2,\kappa \in [0,1]$, it follows that

$$\begin{array}{c}\hfill 2\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)\le \mathrm{u}\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)+\mathrm{u}\left(\frac{2-\kappa }{2}{\mathbf{c}}_1+\frac{\kappa }{2}{\mathbf{c}}_2\right).\end{array}$$

(15)

Multiplying both sides of (15) by ${\kappa }^{\nu -1}w\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)$ and integrating the resulting inequality with respect to $\kappa$ over $[0,1]$, we obtain

$$\begin{array}{c}2\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right){\int }_0^1{\kappa }^{\nu -1}w\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)d\kappa \hfill \\ \le {\int }_0^1{\kappa }^{\nu -1}\mathrm{u}\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)w\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)d\kappa \\ \hfill +{\int }_0^1{\kappa }^{\nu -1}\mathrm{u}\left(\frac{2-\kappa }{2}{\mathbf{c}}_1+\frac{\kappa }{2}{\mathbf{c}}_2\right)w\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)d\kappa .\end{array}$$

(16)

From the left-hand side of the inequality in (16), we use (13) to obtain

$$\begin{array}{cc}\hfill & \frac{2^{\nu -1}\mathsf{\Gamma }\left(\nu \right)}{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu }}\left[\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)+\left({\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)\right]\hfill \\ \hfill & =\frac{2^{\nu }\mathsf{\Gamma }\left(\nu \right)}{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu }}\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\hfill \\ \hfill & =\frac{2^{\nu }}{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu }}{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}(w\circ \varrho )\left(x\right){\varrho }^{\prime }\left(x\right)dx\hfill \\ \hfill & ={\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\left(\frac{2({\mathbf{c}}_2-\varrho \left(x\right))}{{\mathbf{c}}_2-{\mathbf{c}}_1}\right)}^{\nu -1}(w\circ \varrho )\left(x\right){\varrho }^{\prime }\left(x\right)\frac{2dx}{{\mathbf{c}}_2-{\mathbf{c}}_1}\hfill \\ \hfill & ={\int }_0^1{\kappa }^{\nu -1}w\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)d\kappa ,\left[\mathrm{denoting}\kappa :=\frac{2({\mathbf{c}}_2-\varrho \left(x\right))}{{\mathbf{c}}_2-{\mathbf{c}}_1}\right].\hfill \end{array}$$

It follows that

$$\begin{array}{c}2\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right){\int }_0^1{\kappa }^{\nu -1}w\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)d\kappa =\frac{2^{\nu }\mathsf{\Gamma }\left(\nu \right)}{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu }}\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)\hfill \\ \hfill \times \left[\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)+\left({\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)\right].\end{array}$$

(17)

By evaluating the weighted fractional operators, we see that

$$\begin{array}{cc}\hfill & w\left({\mathbf{c}}_2\right)\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}_{w\circ \varrho }^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)+w\left({\mathbf{c}}_1\right)\left({{}_{w\circ \varrho }\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)\hfill \\ \hfill & =w\left({\mathbf{c}}_2\right)\frac{{(w\circ \varrho )}^{-1}\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}(\mathrm{u}\circ \varrho )\left(x\right)(w\circ \varrho )\left(x\right){\varrho }^{\prime }\left(x\right)dx\hfill \\ \hfill & +w\left({\mathbf{c}}_1\right)\frac{{(w\circ \varrho )}^{-1}\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}{(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}(\mathrm{u}\circ \varrho )\left(x\right)(w\circ \varrho )\left(x\right){\varrho }^{\prime }\left(x\right)dx\hfill \\ \hfill & =\frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu }}{2^{\nu }\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\left(\frac{2({\mathbf{c}}_2-\varrho \left(x\right))}{{\mathbf{c}}_2-{\mathbf{c}}_1}\right)}^{\nu -1}(\mathrm{u}\circ \varrho )\left(x\right)(w\circ \varrho )\left(x\right){\varrho }^{\prime }\left(x\right)\frac{2dx}{{\mathbf{c}}_2-{\mathbf{c}}_1}\hfill \\ \hfill & +\frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu }}{2^{\nu }\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}{\left(\frac{2(\varrho \left(x\right)-{\mathbf{c}}_1)}{{\mathbf{c}}_2-{\mathbf{c}}_1}\right)}^{\nu -1}(\mathrm{u}\circ \varrho )\left(x\right)(w\circ \varrho )\left(x\right){\varrho }^{\prime }\left(x\right)\frac{2dx}{{\mathbf{c}}_2-{\mathbf{c}}_1},\hfill \end{array}$$

where we used

$$\begin{array}{cc}\hfill {\left[(w\circ \varrho )\left({\varrho }^{-1}\left(y\right)\right)\right]}^{-1}& =\frac{1}{(w\circ \varrho )\left({\varrho }^{-1}\left(y\right)\right)}=\frac{1}{w\left(y\right)}\mathrm{for}y={\mathbf{c}}_1,{\mathbf{c}}_2.\hfill \end{array}$$

(18)

Setting $t_1=\frac{2({\mathbf{c}}_2-\varrho \left(x\right))}{{\mathbf{c}}_2-{\mathbf{c}}_1}$ and $t_2=\frac{2(\varrho \left(x\right)-{\mathbf{c}}_1)}{{\mathbf{c}}_2-{\mathbf{c}}_1}$, one can deduce that

$$\begin{array}{cc}\hfill & w\left({\mathbf{c}}_2\right)\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}_{w\circ \varrho }^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)+w\left({\mathbf{c}}_1\right)\left({{}_{w\circ \varrho }\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)\hfill \\ \hfill & =\frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu }}{2^{\nu }\mathsf{\Gamma }\left(\nu \right)}[{\int }_0^1{t_1}^{\nu -1}\mathrm{u}\left(\frac{t_1}{2}{\mathbf{c}}_1+\frac{2-t_1}{2}{\mathbf{c}}_2\right)w\left(\frac{t_1}{2}{\mathbf{c}}_1+\frac{2-t_1}{2}{\mathbf{c}}_2\right)d{t}_1\hfill \\ \hfill & +{\int }_0^1{t_2}^{\nu -1}\mathrm{u}\left(\frac{2-t_2}{2}{\mathbf{c}}_1+\frac{t_2}{2}{\mathbf{c}}_2\right)w\left(\frac{2-t_2}{2}{\mathbf{c}}_1+\frac{t_2}{2}{\mathbf{c}}_2\right)d{t}_2\hfill \\ \hfill & =\frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu }}{2^{\nu }\mathsf{\Gamma }\left(\nu \right)}[{\int }_0^1{\kappa }^{\nu -1}\mathrm{u}\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)w\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)d\kappa \hfill \\ \hfill & +{\int }_0^1{\kappa }^{\nu -1}\mathrm{u}\left(\frac{2-\kappa }{2}{\mathbf{c}}_1+\frac{\kappa }{2}{\mathbf{c}}_2\right)\underset{\mathrm{by}\mathrm{using}\left(12\right)}{\underbrace{w\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)}}d\kappa ].\hfill \end{array}$$

It follows that

$$\begin{array}{c}{\int }_0^1{\kappa }^{\nu -1}\mathrm{u}\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)w\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)d\kappa \hfill \\ \hfill +{\int }_0^1{\kappa }^{\nu -1}\mathrm{u}\left(\frac{2-\kappa }{2}{\mathbf{c}}_1+\frac{\kappa }{2}{\mathbf{c}}_2\right)w\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)d\kappa \end{array}$$

$$\begin{array}{c}=\frac{2^{\nu }\mathsf{\Gamma }\left(\nu \right)}{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu }}[w\left({\mathbf{c}}_2\right)\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}_{w\circ \varrho }^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\hfill \\ \hfill +w\left({\mathbf{c}}_1\right)\left({{}_{w\circ \varrho }\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)].\end{array}$$

(19)

By making use of (17) and (19) in (16), we get

$$\begin{array}{c}\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)[\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\hfill \\ +\left({\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)]\le w\left({\mathbf{c}}_2\right)\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}_{w\circ \varrho }^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\\ \hfill +w\left({\mathbf{c}}_1\right)\left({{}_{w\circ \varrho }\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right).\end{array}$$

(20)

Thus, the proof of the first inequality of (14) is completed.

On the other hand, we can prove the second inequality of (14) by making use of the convexity of $\mathrm{u}$ to get

$$\begin{array}{c}\hfill \mathrm{u}\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)+\mathrm{u}\left(\frac{2-\kappa }{2}{\mathbf{c}}_1+\frac{\kappa }{2}{\mathbf{c}}_2\right)\le \mathrm{u}\left({\mathbf{c}}_1\right)+\mathrm{u}\left({\mathbf{c}}_2\right).\end{array}$$

(21)

Multiplying both sides of (21) by ${\kappa }^{\nu -1}w\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)$ and integrating with respect to $\kappa$ over $[0,1]$ to get

$$\begin{array}{c}{\int }_0^1{\kappa }^{\nu -1}\mathrm{u}\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)w\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)d\kappa \hfill \\ +{\int }_0^1{\kappa }^{\nu -1}\mathrm{u}\left(\frac{2-\kappa }{2}{\mathbf{c}}_1+\frac{\kappa }{2}{\mathbf{c}}_2\right)w\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)d\kappa \\ \hfill \le \left(\mathrm{u}\left({\mathbf{c}}_1\right)+\mathrm{u}\left({\mathbf{c}}_2\right)\right){\int }_0^1{\kappa }^{\nu -1}w\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)d\kappa .\end{array}$$

(22)

Then, by using (12) and (19) in (22), we get

$$\begin{array}{c}w\left({\mathbf{c}}_2\right)\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}_{w\circ \varrho }^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\hfill \\ +w\left({\mathbf{c}}_1\right)\left({{}_{w\circ \varrho }\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)\\ \le \frac{\mathrm{u}\left({\mathbf{c}}_1\right)+\mathrm{u}\left({\mathbf{c}}_2\right)}{2}[\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\\ \hfill +\left({\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)].\end{array}$$

This ends our proof. □

Remark 3. 

From Theorem 1, we can obtain some special cases as follows:

(i)

If $\varrho \left(x\right)=x$, then inequality (14) becomes

$$\begin{array}{c}\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)\left[{{}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}{}^{RL}\mathcal{I}}^{\nu }w\left({\mathbf{c}}_2\right)+{{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu }w\left({\mathbf{c}}_1\right)\right]\hfill \\ \le w\left({\mathbf{c}}_2\right)\left({{}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}{}^{RL}\mathcal{I}}_w^{\nu }\mathrm{u}\right)\left({\mathbf{c}}_2\right)+w\left({\mathbf{c}}_1\right)\left({{}_w{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu }\mathrm{u}\right)\left({\mathbf{c}}_1\right)\\ \hfill \le \frac{\mathrm{u}\left({\mathbf{c}}_1\right)+\mathrm{u}\left({\mathbf{c}}_2\right)}{2}\left[{{}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}{}^{RL}\mathcal{I}}^{\nu }w\left({\mathbf{c}}_2\right)+{{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu }w\left({\mathbf{c}}_1\right)\right],\end{array}$$

(23)

where ${{}_{{\mathbf{c}}_1+}{}^{RL}\mathcal{I}}_w^{\nu }$ and ${{}_w{}^{RL}\mathcal{I}}_{{\mathbf{c}}_2-}^{\nu }$ are the left- and right-weighted $\mathtt{RL}$ fractional integrals, respectively, given by

$$\begin{array}{cc}\hfill \left({{}_{{\mathbf{c}}_1+}{}^{RL}\mathcal{I}}_w^{\nu }\mathrm{u}\right)\left(x\right)& =\frac{w^{-1}\left(x\right)}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\mathbf{c}}_1}^x{(x-\kappa )}^{\nu -1}\mathrm{u}\left(\kappa \right)w\left(\kappa \right)d\kappa ,\hfill \\ \hfill \left({{}_w{}^{RL}\mathcal{I}}_{{\mathbf{c}}_2-}^{\nu }\mathrm{u}\right)\left(x\right)& =\frac{w^{-1}\left(x\right)}{\mathsf{\Gamma }\left(\nu \right)}{\int }_x^{{\mathbf{c}}_2}{(\kappa -x)}^{\nu -1}\mathrm{u}\left(\kappa \right)w\left(\kappa \right)d\kappa ,\nu >0.\hfill \end{array}$$

(ii)

If $\varrho \left(x\right)=x$ and $\nu =1$, then inequality (14) becomes the inequality in (7).

(iii)

If $\varrho \left(x\right)=x$ and $w\left(x\right)=1$, then inequality (14) becomes the inequality in (5).

(iv)

If $\varrho \left(x\right)=x$, $w\left(x\right)=1$ and $\nu =1$, then inequality (14) becomes the inequality in (2).

Lemma 2. 

Let $0\le {\mathbf{c}}_1<{\mathbf{c}}_2,$ let $\mathrm{u}:[{\mathbf{c}}_1,{\mathbf{c}}_2]\to \mathcal{R}$ be a continuous with a derivative ${\mathrm{u}}^{\prime }\in L^1[{\mathbf{c}}_1,{\mathbf{c}}_2]$ such that $\mathrm{u}\left(x\right)=\mathrm{u}\left({\mathbf{c}}_1\right)+{\int }_{{\mathbf{c}}_1}^x{\mathrm{u}}^{\prime }\left(\kappa \right)d\kappa$ and let $w:[{\mathbf{c}}_1,{\mathbf{c}}_2]\to \mathcal{R}$ be an integrable, positive and weighted symmetric function with respect to $\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}$. If ϱ is a continuous increasing mapping from the interval $[{\mathbf{c}}_1,{\mathbf{c}}_2)$ onto itself with a derivative ${\varrho }^{\prime }\left(x\right)$ which is continuous on $({\mathbf{c}}_1,{\mathbf{c}}_2)$, then for $\nu >0$, the following equality is valid:

$$\begin{array}{c}\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)[\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\hfill \\ +\left({\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)]\\ -[w\left({\mathbf{c}}_2\right)\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}_{w\circ \varrho }^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\\ +w\left({\mathbf{c}}_1\right)\left({{}_{w\circ \varrho }\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)]\\ =\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\left[{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right]({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right){\varrho }^{\prime }\left(\kappa \right)d\kappa \\ \hfill -\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\left[{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right]({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right){\varrho }^{\prime }\left(\kappa \right)d\kappa .\end{array}$$

(24)

Proof. 

Let us set

$$\begin{array}{cc}\hfill & \frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\left[{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right]({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right){\varrho }^{\prime }\left(\kappa \right)d\kappa \hfill \\ \hfill & -\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\left[{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right]({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right){\varrho }^{\prime }\left(\kappa \right)d\kappa \hfill \\ \hfill & =\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\left[{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right]({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right){\varrho }^{\prime }\left(\kappa \right)d\kappa \hfill \\ \hfill & +\frac{-1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\left[{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right]({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right){\varrho }^{\prime }\left(\kappa \right)d\kappa \hfill \\ \hfill & :={\Xi }_1+{\Xi }_2.\hfill \end{array}$$

By integrating by parts, using Lemma 1, and (10) and (11), we obtain

$$\begin{array}{cc}\hfill {\Xi }_1& =\frac{1}{\mathsf{\Gamma }\left(\nu \right)}\left({\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right)(\mathrm{u}\circ \varrho )\left(\kappa \right)d\kappa {|}_{\kappa ={\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\hfill \\ \hfill & -\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}{\varrho }^{\prime }\left(\kappa \right){(\varrho \left(\kappa \right)-{\mathbf{c}}_1)}^{\nu -1}(w\circ \varrho )\left(\kappa \right)(\mathrm{u}\circ \varrho )\left(\kappa \right)d\kappa \hfill \\ \hfill & =\left(\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right)\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)\hfill \\ \hfill & -\underset{\mathrm{by}\mathrm{using}\left(18\right)}{\underbrace{w\left({\mathbf{c}}_1\right)\frac{{(w\circ \varrho )}^{-1}\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)}{\mathsf{\Gamma }\left(\nu \right)}}}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}{\varrho }^{\prime }\left(\kappa \right){(\varrho \left(\kappa \right)-{\mathbf{c}}_1)}^{\nu -1}(w\circ \varrho )\left(\kappa \right)(\mathrm{u}\circ \varrho )\left(\kappa \right)d\kappa \hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)\left({\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)\hfill \\ \hfill & -w\left({\mathbf{c}}_1\right)\left({{}_{w\circ \varrho }\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)\hfill \\ \hfill & =\frac{1}{2}\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)[\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\hfill \\ \hfill & +\left({\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)]-w\left({\mathbf{c}}_1\right)\left({{}_{w\circ \varrho }\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right).\hfill \end{array}$$

Analogously, we get

$$\begin{array}{cc}\hfill {\Xi }_2& =\frac{-1}{\mathsf{\Gamma }\left(\nu \right)}\left({\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right)(\mathrm{u}\circ \varrho )\left(\kappa \right)d\kappa {|}_{t={\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\hfill \\ \hfill & -\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(\kappa \right){({\mathbf{c}}_2-\varrho \left(\kappa \right))}^{\nu -1}(w\circ \varrho )\left(\kappa \right)(\mathrm{u}\circ \varrho )\left(\kappa \right)d\kappa \hfill \\ \hfill & =\left(\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right)\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)\hfill \\ \hfill & -\underset{\mathrm{by}\mathrm{using}\left(18\right)}{\underbrace{w\left({\mathbf{c}}_2\right)\frac{{(w\circ \varrho )}^{-1}\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)}{\mathsf{\Gamma }\left(\nu \right)}}}{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(\kappa \right){({\mathbf{c}}_2-\varrho \left(\kappa \right))}^{\nu -1}(w\circ \varrho )\left(\kappa \right)(\mathrm{u}\circ \varrho )\left(\kappa \right)d\kappa \hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\hfill \\ \hfill & -w\left({\mathbf{c}}_2\right)\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}_{w\circ \varrho }^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\hfill \\ \hfill & =\frac{1}{2}\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)[\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\hfill \\ \hfill & +\left({\mathcal{I}}_{{\varrho }^{-1}\left({\mathbf{c}}_2\right)-}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)]-w\left({\mathbf{c}}_2\right)\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}_{w\circ \varrho }^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right).\hfill \end{array}$$

Thus, we deduce:

$$\begin{array}{c}{\Xi }_1+{\Xi }_2=\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)[\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\hfill \\ +\left({\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(w\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)]-[w\left({\mathbf{c}}_2\right)\left({{}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}\mathcal{I}}_{w\circ \varrho }^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_2\right)\right)\\ \hfill +w\left({\mathbf{c}}_1\right)\left({{}_{w\circ \varrho }\mathcal{I}}_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu :\varrho }(\mathrm{u}\circ \varrho )\right)\left({\varrho }^{-1}\left({\mathbf{c}}_1\right)\right)],\end{array}$$

which completes the proof of Lemma 2. □

Remark 4. 

From Lemma 2, we can obtain some special cases as follows:

(i)

If $\varrho \left(x\right)=x$, then equality (24) becomes

$$\begin{array}{c}\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)\left[{{}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}{}^{RL}\mathcal{I}}^{\nu }w\left({\mathbf{c}}_2\right)+{{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu }w\left({\mathbf{c}}_1\right)\right]\hfill \\ -\left[w\left({\mathbf{c}}_2\right)\left({{}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}{}^{RL}\mathcal{I}}_w^{\nu }\mathrm{u}\right)\left({\mathbf{c}}_2\right)+w\left({\mathbf{c}}_1\right)\left({{}_w{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu }\mathrm{u}\right)\left({\mathbf{c}}_1\right)\right]\\ =\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\mathbf{c}}_1}^{\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}}\left[{\int }_{{\mathbf{c}}_1}^{\kappa }{(x-{\mathbf{c}}_1)}^{\nu -1}w\left(x\right)dx\right]{\mathrm{u}}^{\prime }\left(\kappa \right)d\kappa \\ \hfill -\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}}^{{\mathbf{c}}_2}\left[{\int }_{\kappa }^{{\mathbf{c}}_2}{({\mathbf{c}}_2-x)}^{\nu -1}w\left(x\right)dx\right]{\mathrm{u}}^{\prime }\left(\kappa \right)d\kappa ,\end{array}$$

(25)

where ${{}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}{}^{RL}\mathcal{I}}_w^{\nu }$ and ${{}_w{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu }$ are as defined in Remark 3.

(ii)

If $\varrho \left(x\right)=x$ and $w\left(x\right)=1$, then equality (24) becomes

$$\begin{array}{c}\frac{2^{\nu -1}\mathsf{\Gamma }(\nu +1)}{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu }}\left[{{}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}{}^{RL}\mathcal{I}}^{\nu }\mathrm{u}\left({\mathbf{c}}_2\right)+{{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu }\mathrm{u}\left({\mathbf{c}}_1\right)\right]-\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)=\frac{{\mathbf{c}}_2-{\mathbf{c}}_1}{4}\hfill \\ \hfill \times \left[{\int }_0^1{\kappa }^{\nu }{\mathrm{u}}^{\prime }\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)d\kappa -{\int }_0^1{\kappa }^{\nu }{\mathrm{u}}^{\prime }\left(\frac{2-\kappa }{2}{\mathbf{c}}_1+\frac{\kappa }{2}{\mathbf{c}}_2\right)d\kappa \right],\end{array}$$

which is already obtained in ([39] [Lemma 3]).

(iii)

If $\varrho \left(x\right)=x,w\left(x\right)=1$ and $\nu =1$, then equality (24) becomes

$$\begin{array}{c}\frac{1}{{\mathbf{c}}_2-{\mathbf{c}}_1}{\int }_{{\mathbf{c}}_1}^{{\mathbf{c}}_2}\mathrm{u}\left(x\right)dx-\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)=\frac{{\mathbf{c}}_2-{\mathbf{c}}_1}{4}[{\int }_0^1\kappa {\mathrm{u}}^{\prime }\left(\frac{\kappa }{2}{\mathbf{c}}_1+\frac{2-\kappa }{2}{\mathbf{c}}_2\right)d\kappa \hfill \\ \hfill -{\int }_0^1\kappa {\mathrm{u}}^{\prime }\left(\frac{2-\kappa }{2}{\mathbf{c}}_1+\frac{\kappa }{2}{\mathbf{c}}_2\right)d\kappa ],\end{array}$$

(26)

which is already obtained in ([39] [Corollary 1]).

3. Main Results

By the help of Lemma 2, we can deduce the following $\mathtt{HHF}$ inequalities.

Theorem 2. 

Let $0\le {\mathbf{c}}_1<{\mathbf{c}}_2$, let $\mathrm{u}:[{\mathbf{c}}_1,{\mathbf{c}}_2]\subseteq [0,\infty )\to \mathcal{R}$ be a (continuously) differentiable function on the interval $[{\mathbf{c}}_1,{\mathbf{c}}_2]$ such that $\mathrm{u}\left(x\right)=\mathrm{u}\left({\mathbf{c}}_1\right)+{\int }_{{\mathbf{c}}_1}^x{\mathrm{u}}^{\prime }\left(\kappa \right)d\kappa$, and let $w:[{\mathbf{c}}_1,{\mathbf{c}}_2]\to \mathcal{R}$ be an integrable, positive and weighted symmetric function with respect to $\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}$. If, in addition, $|{\mathrm{u}}^{\prime }|$ is convex on $[{\mathbf{c}}_1,{\mathbf{c}}_2]$, and ϱ is an increasing and positive function from $[{\mathbf{c}}_1,{\mathbf{c}}_2)$ onto itself such that its derivative ${\varrho }^{\prime }\left(x\right)$ is continuous on $({\mathbf{c}}_1,{\mathbf{c}}_2)$, then for $\nu >0$ the following inequalities are valid:

$$\begin{array}{c}|{\Xi }_1+{\Xi }_2|={\displaystyle |}\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\left[{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right]\hfill \\ \times ({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right){\varrho }^{\prime }\left(\kappa \right)d\kappa \\ \hfill -\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\left[{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right]({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right){\varrho }^{\prime }\left(\kappa \right)d\kappa |\end{array}$$

$$\begin{array}{c}\le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}}{2^{\nu +2}\mathsf{\Gamma }(\nu +3)}\{{\parallel w\parallel }_{\left[{\mathbf{c}}_1,\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right],\infty }\left[(\nu +3)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right)|+(\nu +1)\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|\right]\hfill \\ +{\parallel w\parallel }_{\left[\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2},{\mathbf{c}}_2\right],\infty }\left[(\nu +1)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right)|+(\nu +3)\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|\right]\}\\ \hfill \le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}{\parallel w\parallel }_{\left[{\mathbf{c}}_1,{\mathbf{c}}_2\right],\infty }}{2^{\nu +1}\mathsf{\Gamma }(\nu +2)}\left[|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right)|+|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)|\right].\end{array}$$

(27)

Proof. 

By making use of Lemma 2 and properties of the modulus, we obtain

$$\begin{array}{c}|{\Xi }_1+{\Xi }_2|\hfill \\ =|\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\left[{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right]({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right){\varrho }^{\prime }\left(\kappa \right)d\kappa \\ -\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\left[{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right]({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right){\varrho }^{\prime }\left(\kappa \right)d\kappa |\\ \le \frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\left|{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right|\left|({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right)\right|{\varrho }^{\prime }\left(\kappa \right)d\kappa \\ +\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\left|{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right|\\ \hfill \times |({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right)|{\varrho }^{\prime }\left(\kappa \right)d\kappa .\end{array}$$

(28)

Since $|{\mathrm{u}}^{\prime }|$ is convex on $[{\mathbf{c}}_1,{\mathbf{c}}_2]$, we get for $\kappa \in \left[{\varrho }^{-1}\left({\mathbf{c}}_1\right),{\varrho }^{-1}\left({\mathbf{c}}_2\right)\right]$:

$$\begin{array}{c}\left|({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right)\right|=\left|{\mathrm{u}}^{\prime }\left(\frac{{\mathbf{c}}_2-\varrho \left(\kappa \right)}{{\mathbf{c}}_2-{\mathbf{c}}_1}{\mathbf{c}}_1+\frac{\varrho \left(\kappa \right)-{\mathbf{c}}_1}{{\mathbf{c}}_2-{\mathbf{c}}_1}{\mathbf{c}}_2\right)\right|\hfill \\ \hfill \le \frac{{\mathbf{c}}_2-\varrho \left(\kappa \right)}{{\mathbf{c}}_2-{\mathbf{c}}_1}|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right)|+\frac{\varrho \left(\kappa \right)-{\mathbf{c}}_1}{{\mathbf{c}}_2-{\mathbf{c}}_1}\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|.\end{array}$$

(29)

Hence, we obtain

$$\begin{array}{c}|{\Xi }_1+{\Xi }_2|\le \frac{{\parallel w\parallel }_{\left[{\mathbf{c}}_1,\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right],\infty }}{({\mathbf{c}}_2-{\mathbf{c}}_1)\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\left|{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}dx\right|\hfill \\ \times \left[({\mathbf{c}}_2-\varrho \left(\kappa \right))|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right)|+(\varrho \left(\kappa \right)-{\mathbf{c}}_1)\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|\right]{\varrho }^{\prime }\left(\kappa \right)d\kappa \\ +\frac{{\parallel w\parallel }_{\left[\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2},{\mathbf{c}}_2\right],\infty }}{({\mathbf{c}}_2-{\mathbf{c}}_1)\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\left|{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}dx\right|\\ \times \left[({\mathbf{c}}_2-\varrho \left(\kappa \right))|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right)|+(\varrho \left(\kappa \right)-{\mathbf{c}}_1)\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|\right]{\varrho }^{\prime }\left(\kappa \right)d\kappa \\ =\frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}}{2^{\nu +2}\mathsf{\Gamma }(\nu +3)}\{{\parallel w\parallel }_{\left[{\mathbf{c}}_1,\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right],\infty }\left[(\nu +3)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right)|+(\nu +1)\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|\right]\\ +{\parallel w\parallel }_{\left[\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2},{\mathbf{c}}_2\right],\infty }\left[(\nu +1)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right)|+(\nu +3)\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|\right]\}\\ \hfill \le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}{\parallel w\parallel }_{\left[{\mathbf{c}}_1,{\mathbf{c}}_2\right],\infty }}{2^{\nu +1}\mathsf{\Gamma }(\nu +2)}\left[|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right)|+|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)|\right],\end{array}$$

(30)

where

$$\begin{array}{c}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}dx=\frac{{(\varrho \left(\kappa \right)-{\mathbf{c}}_1)}^{\nu }}{\nu };\hfill \\ \hfill {\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}dx=\frac{{({\mathbf{c}}_2-\varrho \left(\kappa \right))}^{\nu }}{\nu };\end{array}$$

$$\begin{array}{c}\hfill {\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}{(\varrho \left(\kappa \right)-{\mathbf{c}}_1)}^{\nu +1}{\varrho }^{\prime }\left(\kappa \right)d\kappa ={\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{({\mathbf{c}}_2-\varrho \left(\kappa \right))}^{\nu +1}{\varrho }^{\prime }\left(\kappa \right)d\kappa =\frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +2}}{2^{\nu +2}(\nu +2)};\end{array}$$

$$\begin{array}{c}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}{(\varrho \left(\kappa \right)-{\mathbf{c}}_1)}^{\nu }({\mathbf{c}}_2-\varrho \left(\kappa \right)){\varrho }^{\prime }\left(\kappa \right)d\kappa \hfill \\ \hfill ={\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{({\mathbf{c}}_2-\varrho \left(\kappa \right))}^{\nu }(\varrho \left(\kappa \right)-{\mathbf{c}}_1){\varrho }^{\prime }\left(\kappa \right)d\kappa =\frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +2}(\nu +3)}{2^{\nu +2}(\nu +1)(\nu +2)}.\end{array}$$

This completes our proof. □

Remark 5. 

From Theorem 2, we can obtain some special cases as follows:

(i)

If $\varrho \left(x\right)=x$, then inequality (27) becomes

$$\begin{array}{c}|\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)\left[{{}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}{}^{RL}\mathcal{I}}^{\nu }w\left({\mathbf{c}}_2\right)+{{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu }w\left({\mathbf{c}}_1\right)\right]\hfill \\ -\left[w\left({\mathbf{c}}_2\right)\left({{}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}{}^{RL}\mathcal{I}}_w^{\nu }\mathrm{u}\right)\left({\mathbf{c}}_2\right)+w\left({\mathbf{c}}_1\right)\left({{}_w{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu }\mathrm{u}\right)\left({\mathbf{c}}_1\right)\right]|\\ \le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}}{2^{\nu +2}\mathsf{\Gamma }(\nu +3)}\{{\parallel w\parallel }_{\left[{\mathbf{c}}_1,\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right],\infty }\left[(\nu +3)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right)|+(\nu +1)\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|\right]\\ +{\parallel w\parallel }_{\left[\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2},{\mathbf{c}}_2\right],\infty }\left[(\nu +1)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right)|+(\nu +3)\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|\right]\}\\ \hfill \le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}{\parallel w\parallel }_{\left[{\mathbf{c}}_1,{\mathbf{c}}_2\right],\infty }}{2^{\nu +1}\mathsf{\Gamma }(\nu +2)}\left[|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right)|+|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)|\right].\end{array}$$

(31)

(ii)

If $\varrho \left(x\right)=x$ and $w\left(x\right)=1$, then inequality (27) becomes

$$\begin{array}{c}|\frac{2^{\nu -1}\mathsf{\Gamma }(\nu +1)}{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu }}\left[{{}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}{}^{RL}\mathcal{I}}^{\nu }\mathrm{u}\left({\mathbf{c}}_2\right)+{{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu }\mathrm{u}\left({\mathbf{c}}_1\right)\right]-\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)|\hfill \\ \le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}}{2^{\nu +2}\mathsf{\Gamma }(\nu +3)}\{\left[(\nu +3)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right)|+(\nu +1)\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|\right]\\ \hfill +\left[(\nu +1)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right)|+(\nu +3)\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|\right]\}\le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}}{2^{\nu +1}\mathsf{\Gamma }(\nu +2)}\left[|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right)|+|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)|\right],\end{array}$$

(32)

which is already obtained in ([39] [Theorem 5]).

(iii)

If $\varrho \left(x\right)=x,w\left(x\right)=1$ and $\nu =1$, then inequality (27) becomes

$$|\frac{1}{{\mathbf{c}}_2-{\mathbf{c}}_1}{\int }_{{\mathbf{c}}_1}^{{\mathbf{c}}_2}\mathrm{u}\left(x\right)dx-\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)|\le \frac{{\mathbf{c}}_2-{\mathbf{c}}_1}{8}\left[|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right)|+|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)|\right],$$

(33)

which is already obtained in ([45] [Theorem 2.2]).

Theorem 3. 

Let $0\le {\mathbf{c}}_1<{\mathbf{c}}_2$, let $\mathrm{u}:[{\mathbf{c}}_1,{\mathbf{c}}_2]\subseteq [0,\infty )\to \mathcal{R}$ be a (continuously) differentiable function on the interval $[{\mathbf{c}}_1,{\mathbf{c}}_2]$ such that $\mathrm{u}\left(x\right)=\mathrm{u}\left({\mathbf{c}}_1\right)+{\int }_{{\mathbf{c}}_1}^x{\mathrm{u}}^{\prime }\left(\kappa \right)d\kappa$, and let $w:[{\mathbf{c}}_1,{\mathbf{c}}_2]\to \mathcal{R}$ be an integrable, positive and weighted symmetric function with respect to $\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}$. If, in addition, $|{\mathrm{u}}^{\prime }{|}^q$ is convex on $[{\mathbf{c}}_1,{\mathbf{c}}_2]$ with $q\ge 1,$ and ϱ is an increasing and positive function from $[{\mathbf{c}}_1,{\mathbf{c}}_2)$ onto itself such that its derivative ${\varrho }^{\prime }\left(x\right)$ is continuous on $({\mathbf{c}}_1,{\mathbf{c}}_2)$, then for $\nu >0$, the following inequalities are valid:

$$\begin{array}{c}|{\Xi }_1+{\Xi }_2|\le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}}{2^{\nu +1+\frac{1}{q}}{(\nu +2)}^{\frac{1}{q}}\mathsf{\Gamma }(\nu +2)}\hfill \\ \times {\{\parallel w\parallel }_{\left[{\mathbf{c}}_1,\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right],\infty }{\left[(\nu +3)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+(\nu +1){\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\\ +{\parallel w\parallel }_{\left[\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2},{\mathbf{c}}_2\right],\infty }{\left[(\nu +1)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+(\nu +3){\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\}\\ \le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}{\parallel w\parallel }_{\left[{\mathbf{c}}_1,{\mathbf{c}}_2\right],\infty }}{2^{\nu +1+\frac{1}{q}}{(\nu +2)}^{\frac{1}{q}}\mathsf{\Gamma }(\nu +2)}\{{\left[(\nu +3)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+(\nu +1){\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\\ \hfill +{\left[(\nu +1)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+(\nu +3){\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\}.\end{array}$$

(34)

Proof. 

Since $|{\mathrm{u}}^{\prime }{|}^q$ is convex on $[{\mathbf{c}}_1,{\mathbf{c}}_2]$, we get for $\kappa \in \left[{\varrho }^{-1}\left({\mathbf{c}}_1\right),{\varrho }^{-1}\left({\mathbf{c}}_2\right)\right]$:

$$\begin{array}{c}{\left|({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right)\right|}^q={\left|{\mathrm{u}}^{\prime }\left(\frac{{\mathbf{c}}_2-\varrho \left(\kappa \right)}{{\mathbf{c}}_2-{\mathbf{c}}_1}{\mathbf{c}}_1+\frac{\varrho \left(\kappa \right)-{\mathbf{c}}_1}{{\mathbf{c}}_2-{\mathbf{c}}_1}{\mathbf{c}}_2\right)\right|}^q\hfill \\ \hfill \le \frac{{\mathbf{c}}_2-\varrho \left(\kappa \right)}{{\mathbf{c}}_2-{\mathbf{c}}_1}|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+\frac{\varrho \left(\kappa \right)-{\mathbf{c}}_1}{{\mathbf{c}}_2-{\mathbf{c}}_1}{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q.\end{array}$$

(35)

By making use of Lemma 2, power mean inequality and convexity of $|{\mathrm{u}}^{\prime }{|}^q$, we get

$$\begin{array}{c}|{\Xi }_1+{\Xi }_2|\hfill \\ \le \frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\left|{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right|\left|({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right)\right|{\varrho }^{\prime }\left(\kappa \right)d\kappa \\ +\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\left|{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right|\left|({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right)\right|{\varrho }^{\prime }\left(\kappa \right)d\kappa \\ \le \frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\left({\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\left|{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right|{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{1-\frac{1}{q}}\\ \times {\left({\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\left|{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right|{\left|({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right)\right|}^q{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}\\ +\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\left({\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\left|{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right|{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{1-\frac{1}{q}}\\ \hfill \times {\left({\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\left|{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right|{\left|({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right)\right|}^q{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}\end{array}$$

$$\begin{array}{c}\le \frac{{\parallel w\parallel }_{\left[{\mathbf{c}}_1,\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right],\infty }}{\mathsf{\Gamma }\left(\nu \right)}\hfill \\ \times {\left({\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\left|{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}dx\right|{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{1-\frac{1}{q}}\\ \hfill \times {\left({\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\left|{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}dx\right|{\left|({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right)\right|}^q{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}\end{array}$$

$$\begin{array}{c}+\frac{{\parallel w\parallel }_{\left[\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2},{\mathbf{c}}_2\right],\infty }}{\mathsf{\Gamma }\left(\nu \right)}\hfill \\ \times {\left({\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\left|{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}dx\right|{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{1-\frac{1}{q}}\\ \hfill \times {\left({\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\left|{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}dx\right|{\left|({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right)\right|}^q{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}\end{array}$$

$$\begin{array}{c}\le \frac{{\parallel w\parallel }_{\left[{\mathbf{c}}_1,\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right],\infty }}{\mathsf{\Gamma }\left(\nu \right)}\hfill \\ {\left({\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\left|{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}dx\right|{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{1-\frac{1}{q}}\\ \times \left[{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\right|{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}dx|\\ \times \left(\frac{{\mathbf{c}}_2-\varrho \left(\kappa \right)}{{\mathbf{c}}_2-{\mathbf{c}}_1}|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+\frac{\varrho \left(\kappa \right)-{\mathbf{c}}_1}{{\mathbf{c}}_2-{\mathbf{c}}_1}{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right){\varrho }^{\prime }\left(\kappa \right)d\kappa {]}^{\frac{1}{q}}\\ +\frac{{\parallel w\parallel }_{\left[\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2},{\mathbf{c}}_2\right],\infty }}{\mathsf{\Gamma }\left(\nu \right)}{\left({\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\left|{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}dx\right|{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{1-\frac{1}{q}}\\ \times \left[{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\right|{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}dx|\\ \times \left(\frac{{\mathbf{c}}_2-\varrho \left(\kappa \right)}{{\mathbf{c}}_2-{\mathbf{c}}_1}|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+\frac{\varrho \left(\kappa \right)-{\mathbf{c}}_1}{{\mathbf{c}}_2-{\mathbf{c}}_1}{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right){\varrho }^{\prime }\left(\kappa \right)d\kappa {]}^{\frac{1}{q}}\\ =\frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}}{2^{\nu +1+\frac{1}{q}}{(\nu +2)}^{\frac{1}{q}}\mathsf{\Gamma }(\nu +2)}\\ \times \{{\parallel w\parallel }_{\left[{\mathbf{c}}_1,\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right],\infty }{\left[(\nu +3)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+(\nu +1){\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\\ \hfill +{\parallel w\parallel }_{\left[\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2},{\mathbf{c}}_2\right],\infty }{\left[(\nu +1)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+(\nu +3){\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\}\end{array}$$

$$\begin{array}{c}\le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}{\parallel w\parallel }_{\left[{\mathbf{c}}_1,{\mathbf{c}}_2\right],\infty }}{2^{\nu +1+\frac{1}{q}}{(\nu +2)}^{\frac{1}{q}}\mathsf{\Gamma }(\nu +2)}\{{\left[(\nu +3)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+(\nu +1){\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\hfill \\ \hfill +{\left[(\nu +1)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+(\nu +3){\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\},\end{array}$$

(36)

where it is easily seen that

$$\begin{array}{c}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\left|{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}dx\right|{\varrho }^{\prime }\left(\kappa \right)d\kappa \hfill \\ \hfill ={\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\left|{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}dx\right|{\varrho }^{\prime }\left(\kappa \right)d\kappa =\frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}}{2^{\nu +1}\nu (\nu +1)}.\end{array}$$

Hence, the proof is completed. □

Remark 6. 

From Theorem 3, we can obtain some special cases as follows:

(i)

If $\varrho \left(x\right)=x$, then inequality (34) becomes

$$\begin{array}{c}|\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)\left[{{}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}{}^{RL}\mathcal{I}}^{\nu }w\left({\mathbf{c}}_2\right)+{{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu }w\left({\mathbf{c}}_1\right)\right]\hfill \\ \hfill -\left[w\left({\mathbf{c}}_2\right)\left({{}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}{}^{RL}\mathcal{I}}_w^{\nu }\mathrm{u}\right)\left({\mathbf{c}}_2\right)+w\left({\mathbf{c}}_1\right)\left({{}_w{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu }\mathrm{u}\right)\left({\mathbf{c}}_1\right)\right]|\end{array}$$

$$\begin{array}{c}\le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}}{2^{\nu +1+\frac{1}{q}}{(\nu +2)}^{\frac{1}{q}}\mathsf{\Gamma }(\nu +2)}\hfill \\ \times {\{\parallel w\parallel }_{\left[{\mathbf{c}}_1,\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right],\infty }{\left[(\nu +3)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+(\nu +1){\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\\ +{\parallel w\parallel }_{\left[\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2},{\mathbf{c}}_2\right],\infty }{\left[(\nu +1)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+(\nu +3){\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\}\\ \le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}{\parallel w\parallel }_{\left[{\mathbf{c}}_1,{\mathbf{c}}_2\right],\infty }}{2^{\nu +1+\frac{1}{q}}{(\nu +2)}^{\frac{1}{q}}\mathsf{\Gamma }(\nu +2)}\{{\left[(\nu +3)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+(\nu +1){\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\\ \hfill +{\left[(\nu +1)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+(\nu +3){\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\}.\end{array}$$

(37)

(ii)

If $\varrho \left(x\right)=x$ and $w\left(x\right)=1$, then inequality (34) becomes

$$\begin{array}{c}|\frac{2^{\nu -1}\mathsf{\Gamma }(\nu +1)}{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu }}\left[{{}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}{}^{RL}\mathcal{I}}^{\nu }\mathrm{u}\left({\mathbf{c}}_2\right)+{{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu }\mathrm{u}\left({\mathbf{c}}_1\right)\right]-\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)|\hfill \\ \le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}}{2^{\nu +1+\frac{1}{q}}{(\nu +2)}^{\frac{1}{q}}\mathsf{\Gamma }(\nu +2)}\{{\left[(\nu +3)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+(\nu +1){\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\\ \hfill +{\left[(\nu +1)|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+(\nu +3){\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\},\end{array}$$

(38)

which is already obtained in ([39] [Theorem 5]).

(iii)

If $\varrho \left(x\right)=x,w\left(x\right)=1$ and $\nu =1$, then inequality (34) becomes

$$\begin{array}{c}|\frac{1}{{\mathbf{c}}_2-{\mathbf{c}}_1}{\int }_{{\mathbf{c}}_1}^{{\mathbf{c}}_2}\mathrm{u}\left(x\right)dx-\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)|\hfill \\ \hfill \le \frac{{\mathbf{c}}_2-{\mathbf{c}}_1}{8\sqrt[q]{3}}\left\{{\left[|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+2{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}+{\left[2|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\right\}.\end{array}$$

(39)

Theorem 4. 

Let $0\le {\mathbf{c}}_1<{\mathbf{c}}_2$, let $\mathrm{u}:[{\mathbf{c}}_1,{\mathbf{c}}_2]\subseteq [0,\infty )\to \mathcal{R}$ be a (continuously) differentiable function on the interval $[{\mathbf{c}}_1,{\mathbf{c}}_2]$ such that $\mathrm{u}\left(x\right)=\mathrm{u}\left({\mathbf{c}}_1\right)+{\int }_{{\mathbf{c}}_1}^x{\mathrm{u}}^{\prime }\left(\kappa \right)d\kappa$, and let $w:[{\mathbf{c}}_1,{\mathbf{c}}_2]\to \mathcal{R}$ be an integrable, positive and weighted symmetric function with respect to $\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}$. If, in addition, $|{\mathrm{u}}^{\prime }{|}^q$ is convex on $[{\mathbf{c}}_1,{\mathbf{c}}_2]$ with $\frac{1}{p}+\frac{1}{q}=1$ and $q>1,$ and ϱ is an increasing and positive function from $[{\mathbf{c}}_1,{\mathbf{c}}_2)$ onto itself such that its derivative ${\varrho }^{\prime }\left(x\right)$ is continuous on $({\mathbf{c}}_1,{\mathbf{c}}_2)$, then for $\nu >0$ the following inequalities are valid:

$$\begin{array}{c}|{\Xi }_1+{\Xi }_2|\le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}}{2^{\nu +1+\frac{2}{q}}{(p\nu +1)}^{\frac{1}{p}}\mathsf{\Gamma }(\nu +1)}\{{\parallel w\parallel }_{\left[{\mathbf{c}}_1,\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right],\infty }\hfill \\ \times {\left[3|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}+{\parallel w\parallel }_{\left[\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2},{\mathbf{c}}_2\right],\infty }{\left[|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+3{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\}\\ \le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}{\parallel w\parallel }_{\left[{\mathbf{c}}_1,{\mathbf{c}}_2\right],\infty }}{2^{\nu +1+\frac{2}{q}}{(p\nu +1)}^{\frac{1}{p}}\mathsf{\Gamma }(\nu +1)}\\ \hfill \times \left\{{\left[3|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}+{\left[|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+3{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\right\}.\end{array}$$

(40)

Proof. 

Since $|{\mathrm{u}}^{\prime }{|}^q$ is convex on $[{\mathbf{c}}_1,{\mathbf{c}}_2]$, we get for $\kappa \in \left[{\varrho }^{-1}\left({\mathbf{c}}_1\right),{\varrho }^{-1}\left({\mathbf{c}}_2\right)\right]$:

$$\begin{array}{c}{\left|({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right)\right|}^q={\left|{\mathrm{u}}^{\prime }\left(\frac{{\mathbf{c}}_2-\varrho \left(\kappa \right)}{{\mathbf{c}}_2-{\mathbf{c}}_1}{\mathbf{c}}_1+\frac{\varrho \left(\kappa \right)-{\mathbf{c}}_1}{{\mathbf{c}}_2-{\mathbf{c}}_1}{\mathbf{c}}_2\right)\right|}^q\hfill \\ \hfill \le \frac{{\mathbf{c}}_2-\varrho \left(\kappa \right)}{{\mathbf{c}}_2-{\mathbf{c}}_1}|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+\frac{\varrho \left(\kappa \right)-{\mathbf{c}}_1}{{\mathbf{c}}_2-{\mathbf{c}}_1}{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q.\end{array}$$

By using Lemma 2, Hölder’s inequality, convexity of $|{\mathrm{u}}^{\prime }{|}^q$ and properties of modulus, we have

$$\begin{array}{c}|{\Xi }_1+{\Xi }_2|\le \frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\left|{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right|\hfill \\ \times |({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right)|{\varrho }^{\prime }\left(\kappa \right)d\kappa \\ +\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\left|{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}(w\circ \varrho )\left(x\right)dx\right|\left|({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right)\right|{\varrho }^{\prime }\left(\kappa \right)d\kappa \\ \hfill \le \frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\left({\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}|{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}(w\circ \varrho )\left(x\right)dx{|}^p{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{p}}\end{array}$$

$$\begin{array}{c}\times {\left({\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}{\left|({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right)\right|}^q{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}\hfill \\ +\frac{1}{\mathsf{\Gamma }\left(\nu \right)}{\left({\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}|{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}(w\circ \varrho )\left(x\right)dx{|}^p{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{p}}\\ \hfill \times {\left({\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\left|({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right)\right|}^q{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}\end{array}$$

$$\begin{array}{c}\le \frac{{\parallel w\parallel }_{\left[{\mathbf{c}}_1,\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right],\infty }}{\mathsf{\Gamma }\left(\nu \right)}\hfill \\ \times {\left({\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}|{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}dx{|}^p{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{p}}\\ \times {\left({\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}{\left|({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right)\right|}^q{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}+\frac{{\parallel w\parallel }_{\left[\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2},{\mathbf{c}}_2\right],\infty }}{\mathsf{\Gamma }\left(\nu \right)}\\ \times {\left({\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}|{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}dx{|}^p{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{p}}\\ \times {\left({\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\left|({\mathrm{u}}^{\prime }\circ \varrho )\left(\kappa \right)\right|}^q{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{q}}\\ \le \frac{{\parallel w\parallel }_{\left[{\mathbf{c}}_1,\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right],\infty }}{\mathsf{\Gamma }\left(\nu \right)}\\ \times {\left({\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}|{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}dx{|}^p{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{p}}\\ \times {\left[{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}\left(\frac{{\mathbf{c}}_2-\varrho \left(\kappa \right)}{{\mathbf{c}}_2-{\mathbf{c}}_1}|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+\frac{\varrho \left(\kappa \right)-{\mathbf{c}}_1}{{\mathbf{c}}_2-{\mathbf{c}}_1}{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right){\varrho }^{\prime }\left(\kappa \right)d\kappa \right]}^{\frac{1}{q}}\\ +\frac{{\parallel w\parallel }_{\left[\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2},{\mathbf{c}}_2\right],\infty }}{\mathsf{\Gamma }\left(\nu \right)}\\ \times {\left({\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}|{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}dx{|}^p{\varrho }^{\prime }\left(\kappa \right)d\kappa \right)}^{\frac{1}{p}}\\ \hfill \times {\left[{\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}\left(\frac{{\mathbf{c}}_2-\varrho \left(\kappa \right)}{{\mathbf{c}}_2-{\mathbf{c}}_1}|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+\frac{\varrho \left(\kappa \right)-{\mathbf{c}}_1}{{\mathbf{c}}_2-{\mathbf{c}}_1}{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right){\varrho }^{\prime }\left(\kappa \right)d\kappa \right]}^{\frac{1}{q}}\end{array}$$

$$\begin{array}{c}=\frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}}{2^{\nu +1+\frac{2}{q}}{(p\nu +1)}^{\frac{1}{p}}\mathsf{\Gamma }(\nu +1)}\{{\parallel w\parallel }_{\left[{\mathbf{c}}_1,\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right],\infty }{\left[3|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\hfill \\ +{\parallel w\parallel }_{\left[\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2},{\mathbf{c}}_2\right],\infty }{\left[|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+3{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\\ \le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}{\parallel w\parallel }_{\left[{\mathbf{c}}_1,{\mathbf{c}}_2\right],\infty }}{2^{\nu +1+\frac{2}{q}}{(p\nu +1)}^{\frac{1}{p}}\mathsf{\Gamma }(\nu +1)}\\ \hfill \times \left\{{\left[3|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}+{\left[|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+3{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\right\},\end{array}$$

where we used the identity

$$\begin{array}{c}{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}|{\int }_{{\varrho }^{-1}\left({\mathbf{c}}_1\right)}^{\kappa }{\varrho }^{\prime }\left(x\right){(\varrho \left(x\right)-{\mathbf{c}}_1)}^{\nu -1}dx{|}^p{\varrho }^{\prime }\left(\kappa \right)d\kappa \hfill \\ \hfill ={\int }_{{\varrho }^{-1}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)}^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}|{\int }_{\kappa }^{{\varrho }^{-1}\left({\mathbf{c}}_2\right)}{\varrho }^{\prime }\left(x\right){({\mathbf{c}}_2-\varrho \left(x\right))}^{\nu -1}dx{|}^p{\varrho }^{\prime }\left(\kappa \right)d\kappa =\frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{p\nu +1}}{2^{p\nu +1}(p\nu +1){\nu }^p}.\end{array}$$

This ends our proof. □

Remark 7. 

From Theorem 4, we can obtain some special cases as follows:

(i)

If $\varrho \left(x\right)=x$, then inequality (40) becomes

$$\begin{array}{c}|\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)\left[{{}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}{}^{RL}\mathcal{I}}^{\nu }w\left({\mathbf{c}}_2\right)+{{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu }w\left({\mathbf{c}}_1\right)\right]\hfill \\ -\left[w\left({\mathbf{c}}_2\right)\left({{}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}{}^{RL}\mathcal{I}}_w^{\nu }\mathrm{u}\right)\left({\mathbf{c}}_2\right)+w\left({\mathbf{c}}_1\right)\left({{}_w{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu }\mathrm{u}\right)\left({\mathbf{c}}_1\right)\right]|\\ \le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}}{2^{\nu +1+\frac{2}{q}}{(p\nu +1)}^{\frac{1}{p}}\mathsf{\Gamma }(\nu +1)}\{{\parallel w\parallel }_{\left[{\mathbf{c}}_1,\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right],\infty }{\left[3|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\\ +{\parallel w\parallel }_{\left[\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2},{\mathbf{c}}_2\right],\infty }{\left[|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+3{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\}\le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}{\parallel w\parallel }_{\left[{\mathbf{c}}_1,{\mathbf{c}}_2\right],\infty }}{2^{\nu +1+\frac{2}{q}}{(p\nu +1)}^{\frac{1}{p}}\mathsf{\Gamma }(\nu +1)}\\ \hfill \times \left\{{\left[3|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}+{\left[|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+3{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\right\}.\end{array}$$

(ii)

If $\varrho \left(x\right)=x$ and $w\left(x\right)=1$, then inequality (40) becomes

$$\begin{array}{c}|\frac{2^{\nu -1}\mathsf{\Gamma }(\nu +1)}{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu }}\left[{{}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+}{}^{RL}\mathcal{I}}^{\nu }\mathrm{u}\left({\mathbf{c}}_2\right)+{{}^{RL}\mathcal{I}}_{\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)-}^{\nu }\mathrm{u}\left({\mathbf{c}}_1\right)\right]-\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)|\hfill \\ \le \frac{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu +1}{\parallel w\parallel }_{\left[{\mathbf{c}}_1,{\mathbf{c}}_2\right],\infty }}{2^{\nu +1+\frac{2}{q}}{(p\nu +1)}^{\frac{1}{p}}\mathsf{\Gamma }(\nu +1)}\\ \hfill \times \left\{{\left[3|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}+{\left[|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+3{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\right\},\end{array}$$

which is already obtained in ([39] [Theorem 6]).

(iii)

If $\varrho \left(x\right)=x,w\left(x\right)=1$ and $\nu =1$, then inequality (40) becomes

$$\begin{array}{c}|\frac{1}{{\mathbf{c}}_2-{\mathbf{c}}_1}{\int }_{{\mathbf{c}}_1}^{{\mathbf{c}}_2}\mathrm{u}\left(x\right)dx-\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)|\le \frac{{\mathbf{c}}_2-{\mathbf{c}}_1}{16}{\left(\frac{4}{p+1}\right)}^{\frac{1}{p}}\hfill \\ \hfill \times \left\{{\left[3|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}+{\left[|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_1\right){|}^q+3{\left|{\mathrm{u}}^{\prime }\left({\mathbf{c}}_2\right)\right|}^q\right]}^{\frac{1}{q}}\right\},\end{array}$$

which is already obtained in ([45] [Theorem 2.3]).

4. Concluding Remarks

In the present article, we have investigated a midpoint fractional $\mathtt{HHF}$ integral inequality by using the weighted fractional integrals with positive weighted symmetric function kernels, which is also the midpoint version of (9). Moreover, we have investigated some related results.

The existing versions of $\mathtt{HHF}$ integral inequalities (7) and (8) have been successfully applied to other classes of convex functions, see [46,47,48]. Therefore, our present results can be applied to those classes of convex functions as well.

Furthermore, one can observe that our results in this article are very generic and can be extended to give further potentially useful and interesting $\mathtt{HHF}$ integral inequalities of end-midpoint version, like the following one

$$\begin{array}{c}\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)\le \frac{2^{\nu -1}\mathsf{\Gamma }(\nu +1)}{{({\mathbf{c}}_2-{\mathbf{c}}_1)}^{\nu }}\left[{{}^{RL}\mathcal{I}}_{{\mathbf{c}}_1+}^{\nu }\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)+{{}^{RL}\mathcal{I}}_{{\mathbf{c}}_2-}^{\nu }\mathrm{u}\left(\frac{{\mathbf{c}}_1+{\mathbf{c}}_2}{2}\right)\right]\hfill \\ \hfill \le \frac{\mathrm{u}\left({\mathbf{c}}_1\right)+\mathrm{u}\left({\mathbf{c}}_2\right)}{2},\end{array}$$

which was already established by Mohammed and Brevik in [49].