(p,h)-Convex Functions Associated with Hadamard and Fejér-Hadamard Inequalities via k-Fractional Integral Operators

Abstract

In this article, generalized versions of the $k$-fractional Hadamard and Fejér-Hadamard inequalities are constructed. To obtain the generalized versions of these inequalities, $k$-fractional integral operators including the well-known Mittag-Leffler function are utilized. The class of ($p,h$)-convex functions for Hadamard-type inequalities give the generalizations of results which have been proved in literature for $p$-convex, $h$-convex, and several functions deducible from these two classes.

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1. Introduction

The subject of fractional calculus which includes the study of fractional-order integrals and derivatives has become a very prominent research topic in recent years. It has got the attention of researchers working in almost all fields of science and engineering. Integral operators are a very important part of fractional calculus; they play an important role in the mathematical treatment of different kinds of real-world problems. In the theory of mathematical inequalities, fractional integral operators appeared as a tool for the generalizations of classical inequalities. In the last two decades, a lot of such generalizations have been published by several authors.

Iscan and Wu [1] have generalized Hermite-Hadamard inequalities for harmonically convex functions by applying fractional integrals. Akkurt et al. [2] have obtained inequalities for $k-h$-Riemann-Liouville fractional integrals with the aid of synchronous and monotonic functions. Mubeen and Habibullah [3] have introduced $k$-analogue of Riemann-Liouville fractional integrals which are used for establishing $k$-fractional versions of well-known inequalities. A study of $k$-fractional integrals has been presented by Tunç et al. in [4], and fractional inequalities have been proved. Farid et al. in [5] demonstrated some novel Fejér-Hadamard and Hadamard inequalities for harmonically convex functions by using fractional integrals containing the Mittag-Leffler function.

Kunt and Iscan [6] presented Hermite-Hadamard-Fejér-type inequalities for $p$-convex functions using fractional integrals. Rashid et al. [7] demonstrated the Hadamard and the Fejér-Hadamard-type inequalities for the generalized fractional integral operator involving Mittag-Leffler functions for preinvexity and $m$-preinvexity. Klnç et al. [8] proved Hadamard and Fejér-Hadamard inequalities for ($h,m$)-strongly convex functions via generalized fractional integrals with Mittag-Leffler functions. Jia et al. [9] demonstrated new types of Hadamard and Fejér-Hadamard fractional integral inequalities for Riemann-Liouville fractional integrals. Yussouf et al. [10] presented generalized types of Hadamard and Fejér-Hadamard-type fractional integral inequalities by utilizing generalized fractional integrals including Mittag-Leffler functions. Faisal et al. [11] proved new Hermite-Hadamard-Jensen-Mercer-type inequalities for convex functions by using Riemann-Liouville fractional integrals. The other recent studies in this area were presented by Zhao et al. [12–14] and Chu et al. [15].

The purpose of this article is to investigate the Hadamard and the Fejér-Hadamard-type inequalities for ($p,h$)-convex functions via generalized $k$-fractional integrals including Mittag-Leffler functions. Inequalities for several kinds of convex functions are special cases of results of this paper. Also, several new inequalities can be deduced from presented results in specific settings. First, we give definitions of integral operators and some results which will set the foundation for this article.

Definition 1 (see [16]).

Let $w,\alpha ,l,\gamma ,c,\in ℂ$, and $\mathfrak{R}l,\mathfrak{R}\alpha >0,\mathfrak{R}c>\mathfrak{R}\gamma >0$ with $\tilde{p}\ge 0,\delta ,\mu >0,\text{and}0<v\le \delta +\mu$. Also, let $\sigma \in L_1\epsilon ,v\text{and}\zeta \in \epsilon ,v$. In that case, the generalized fractional operators $F_{\mu ,\alpha ,l,w,\epsilon }^{\gamma ,\delta ,v,c}+\sigma$ and $F_{\mu ,\alpha ,l,w,v}^{\gamma ,\delta ,v,c}-\sigma$ are defined by $$\begin{array}{c}\text{(1)}& F_{\mu ,\alpha ,l,w,\epsilon +}^{\gamma ,\delta ,v,c}\sigma \zeta ;\tilde{p}={\int }_{\epsilon }^{\zeta }{\zeta -t}^{\alpha -1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{\zeta -t}^{\mu };\tilde{p}\sigma tdt,F_{\mu ,\alpha ,l,w,\upsilon -}^{\gamma ,\delta ,v,c}\sigma \zeta ;\tilde{p}={\int }_{\zeta }^{\upsilon }{t-\zeta }^{\alpha -1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t-\zeta }^{\mu };\tilde{p}\sigma tdt,\end{array}$$where $$\begin{array}{}\text{(2)}& E_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}t;\tilde{p}=\sum _{n=0}^{\infty }\frac{{\beta }_{\tilde{p}}\gamma +nv,c-\gamma }{\beta \gamma ,c-\gamma }\frac{c_{nv}}{\Gamma \mu n+\alpha }\frac{t^n}{l_{n\delta }}\end{array}$$is the enlarged generalized Mittag-Leffler function and ${\beta }_{\tilde{p}}$ is the expansion of beta function described as noted below: $$\begin{array}{}\text{(3)}& {\beta }_{\tilde{p}}\zeta ,\eta ={\int }_0^1{t}^{\zeta -1}{1-t}^{\eta -1}{e}^{-\tilde{p}/t1-t}dt,\end{array}$$where $\mathfrak{R}\zeta ,\mathfrak{R}\eta ,\mathfrak{R}c\tilde{p}>0$.

Definition 2 (see [16]).

Let $\sigma ,\varsigma :\epsilon ,\upsilon ⟶ℝ$ with $0<\epsilon <\upsilon$ be the functions such that $\sigma$ is positive and $\sigma \in L_1\epsilon ,\upsilon$ and $\varsigma$ be differentiable and absolutely increasing. Also, let $\varphi /\zeta$ be an increasing function on $\epsilon ,\infty$, and $\gamma ,c,w,\alpha ,l,\in ℂ$, $\mathfrak{R}l,\mathfrak{R}\alpha >0,\mathfrak{R}c>\mathfrak{R}\gamma >0$ with $\tilde{p}\ge 0,\delta ,\mu >0$, and $0<v\le \mu +\delta$. In that case, for $\zeta \in \epsilon ,\upsilon$, the fractional operators are defined by $$\begin{array}{c}\text{(4)}& {}_{\varsigma }{F}_{\mu ,\alpha ,l,w,\epsilon +}^{\varphi ,\gamma ,\delta ,v,c}\sigma \zeta ;\tilde{p}={\int }_{\epsilon }^{\zeta }\frac{\varphi \varsigma \zeta -\varsigma t}{\varsigma \zeta -\varsigma t}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{\varsigma \zeta -\varsigma t}^{\mu };\tilde{p}{\varsigma }^{\prime }t\sigma tdt,{}_{\varsigma }{F}_{\mu ,\alpha ,l,w,\upsilon -}^{\varphi ,\gamma ,\delta ,v,c}\sigma \zeta ;\tilde{p}={\int }_{\zeta }^{\upsilon }\frac{\varphi \varsigma t-\varsigma \zeta }{\varsigma t-\varsigma \zeta }{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{\varsigma t-\varsigma \zeta }^{\mu };\tilde{p}{\varsigma }^{\prime }t\sigma tdt.\end{array}$$

Definition 3 (see [17]).

Let $\sigma ,\varsigma :\epsilon ,\upsilon ⟶ℝ$ with $0<\epsilon <\upsilon$ be the functions such that $\sigma$ is positive and $\sigma \in L_1\epsilon ,\upsilon$ and $\varsigma$ be differentiable and absolutely increasing. Also, let $w,\gamma ,c,\alpha ,l,\in ℂ$and$\mathfrak{R}l,\mathfrak{R}\alpha >0,\mathfrak{R}c>\mathfrak{R}\gamma >0$with$\tilde{p}\ge 0,\delta ,\mu >0$and$0<v\le \mu +\delta$. In that case, for $\zeta \in \epsilon ,\upsilon$, the unified integral operators are defined by $$\begin{array}{c}\text{(5)}& {}_{\varsigma }{F}_{\mu ,\alpha ,l,w,\epsilon +}^{\gamma ,\delta ,v,c}\sigma \zeta ;\tilde{p}={\int }_{\epsilon }^{\zeta }{\varsigma \zeta -\varsigma t}^{\alpha -1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{\varsigma \zeta -\varsigma t}^{\mu };\tilde{p}{\varsigma }^{\prime }t\sigma tdt,{}_{\varsigma }{F}_{\mu ,\alpha ,l,w,\upsilon -}^{\gamma ,\delta ,v,c}\sigma \zeta ;\tilde{p}={\int }_{\zeta }^{\upsilon }{\varsigma t-\varsigma \zeta }^{\alpha -1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{\varsigma t-\varsigma \zeta }^{\mu };\tilde{p}{\varsigma }^{\prime }t\sigma tdt.\end{array}$$

Recently, Yue et al. [18] described generalized $k$-fractional operators including the further extension of the Mittag-Leffler function as noted below.

Definition 4.

Let $\sigma ,\varsigma :\epsilon ,\upsilon ⟶ℝ$ with $0<\epsilon <\upsilon$ be the functions such that $\sigma$ be positive and $\sigma \in L_1\epsilon ,\upsilon$ and $\varsigma$ be differentiable and absolutely increasing. Let $\alpha >k$ and $\gamma ,c,w,\alpha ,l,\in ℝ,\alpha ,l>0$, and $c>\gamma >0$, with $\tilde{p}\ge 0,\delta ,\mu >0$and$0<v\le \mu +\delta$. In that case, for $\zeta \in \epsilon ,\upsilon$, the left-right generalized $k$-fractional operators ${}_{\varsigma }^k{F}_{\mu ,\alpha ,l,w,\epsilon +}^{\gamma ,\delta ,v,c}\sigma$ and ${}_{\varsigma }^k{F}_{\mu ,\alpha ,l,w,\upsilon -}^{\gamma ,\delta ,v,c}\sigma$ are defined by $$\begin{array}{}\text{(6)}& {}_{\varsigma }^k{F}_{\mu ,\alpha ,l,w,\epsilon +}^{\gamma ,\delta ,v,c}\sigma \zeta ;\tilde{p}={\int }_{\epsilon }^{\zeta }{\varsigma \zeta -\varsigma t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{\varsigma \epsilon -\varsigma t}^{\mu };\tilde{p}{\varsigma }^{\prime }t\sigma tdt,\text{(7)}& {}_{\varsigma }^k{F}_{\mu ,\alpha ,l,w,\upsilon -}^{\gamma ,\delta ,v,c}\sigma \zeta ;\tilde{p}={\int }_{\zeta }^{\upsilon }{\varsigma t-\varsigma \zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{\varsigma t-\varsigma \zeta }^{\mu };\tilde{p}{\varsigma }^{\prime }t\sigma tdt.\end{array}$$

Next, we define convex and related functions.

Definition 5 (see [19]).

A function $\sigma :\epsilon ,\upsilon ⟶ℝ$ is said to be convex, if $$\begin{array}{}\text{(8)}& \sigma t\zeta +1-t\eta \le t\sigma \zeta +1-t\sigma \eta ,\end{array}$$for all $\zeta ,\eta \in \epsilon ,\upsilon$ and $t\in 0,1$.

Definition 6 (see [20]).

Let $I\subset 0,\infty$ be a real interval and $p\in ℝ\backslash 0$. A function $\sigma :I⟶ℝ$ is said to be a $p$-convex function, if $$\begin{array}{}\text{(9)}& \sigma {t{\zeta }^p+1-t{\eta }^p}^{1/p}\le t\sigma \zeta +1-t\sigma \eta ,\end{array}$$for all $\zeta ,\eta \in I$ and $t\in 0,1$.

Definition 7 (see [6]).

Let $p\in ℝ\backslash 0$. In that case, a function $\sigma :\epsilon ,\upsilon \subset 0,\infty ⟶ℝ$ is called $p$-symmetric in accordance with ${{\epsilon }^p+{\upsilon }^p/2}^{1/p}$, if $$\begin{array}{}\text{(10)}& \sigma t^{1/p}=\sigma {{\epsilon }^p-{\upsilon }^p-t}^{1/p},\end{array}$$for $t\in 0,1$.

Definition 8 (see [21]).

Let $h:J⟶ℝ$ be a nonnegative and nonzero function and $p\in ℝ\backslash 0$. We say that $\sigma :I⟶ℝ$ is a $p,h$-convex function, if $\sigma$ is nonnegative and $$\begin{array}{}\text{(11)}& \sigma {t{\zeta }^p+1-t{\eta }^p}^{1/p}\le ht\sigma \zeta +h1-t\sigma \eta ,\end{array}$$for all $\zeta ,\eta \in I$ and $t\in 0,1$. If the above inequality is reversed, then $\sigma$ is said to be a $p,h$-concave function.

Remark 9.

(i) By taking $ht=t$ in Definition 8, we obtain the definition of $p$-convex function

(ii) By taking $ht=t$ and $p=1$ in Definition 8, we obtain the definition of convex function

(iii) By taking $ht=t^s$ and $p=1$ in Definition 8, we obtain the definition of $s$-convex function

(iv) By taking $ht=t^{-1}$ and $p=1$ in Definition 8, we obtain the definition of Godunova-Levin-type convex function

(v) By taking $p=1$ in Definition 8, we obtain the definition of $h$-convex function

(vi) By taking $ht=1$ and $p=1$ in Definition 8, we obtain the definition of $p$-function

The following inequality is the well-known Hadamard inequality.

Theorem 10 (see [22]).

Let $\sigma :\epsilon ,\upsilon ⟶ℝ$ be a convex function for $\epsilon <\upsilon$. Then, the following inequality holds: $$\begin{array}{}\text{(12)}& \sigma \frac{\epsilon +\upsilon }{2}\le \frac{1}{\upsilon -\epsilon }{\int }_{\epsilon }^{\upsilon }\sigma \zeta d\zeta \le \frac{\sigma \epsilon +\sigma \upsilon }{2}.\end{array}$$

The Fejér-Hadamard inequality is a weighted type of the Hadamard inequality presented by Fejér in [23].

Theorem 11.

Let $\sigma :\epsilon ,\upsilon ⟶ℝ$ be a convex function and $\varsigma :\epsilon ,\upsilon ⟶ℝ$ be nonnegative, integrable, and symmetric with respect to $\epsilon +\upsilon /2$. In that case, the below inequality takes: $$\begin{array}{}\text{(13)}& \sigma \frac{\epsilon +\upsilon }{2}{\int }_{\epsilon }^{\upsilon }\varsigma \zeta d\zeta \le {\int }_{\epsilon }^{\upsilon }\sigma \zeta \varsigma \zeta d\zeta \le \frac{\sigma \epsilon +\sigma \upsilon }{2}{\int }_{\epsilon }^{\upsilon }\varsigma \zeta d\zeta .\end{array}$$

For other classes of functions defined after motivating from convex function, the above inequalities have been studied extensively (see [1, 5, 6, 24–26]).

In the upcoming section, first, we construct the Hadamard inequality for $p,h$-convex function via generalized $k$-fractional integrals. Then, an identity is used to construct the Fejér-Hadamard inequality for $p,h$-convex function via generalized $k$-fractional integrals. After, another version of the Hadamard inequality and the Fejér-Hadamard inequality is presented. Moreover, the presented results generalize many already published results.

2. $k$-Fractional Integral Inequalities of Hadamard and Fejér-Hadamard Type

The generalized $k$-fractional Hadamard inequality is stated and proved in the following theorem.

Theorem 12.

Let $h:J⟶ℝ$ be nonnegative, nonzero, and integrable function. Also, let $\sigma ,\varsigma :\epsilon ,\upsilon ⟶ℝ$ with $0<\epsilon <\upsilon$ be the functions such that $\sigma$ is positive and $\sigma \in L_1\epsilon ,\upsilon$, and $\varsigma$ is differentiable and absolutely increasing. If $\sigma$ is $p,h$-convex, and $p\in R\backslash 0$, in that case, the below inequalities for $k$-fractional operators (6) and (7) occur:

(i) If $p>0$, in that case,

$$\begin{array}{}\text{(14)}& \sigma {\frac{{\varsigma }^p\epsilon +{\varsigma }^p\upsilon }{2}}^{1/p}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\epsilon +}^{\gamma ,\delta ,v,c}1{\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}\le h\frac{1}{2}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\epsilon +}^{\gamma ,\delta ,v,c}\sigma \circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}+{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\upsilon -}^{\gamma ,\delta ,v,c}\sigma \circ \theta {\varsigma }^{-1}{\varsigma }^p\epsilon ;\tilde{p}\le h\frac{1}{2}\sigma \varsigma \epsilon +\sigma \varsigma \upsilon {}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},1-}^{\gamma ,\delta ,v,c}h0;\tilde{p}+{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},0+}^{\gamma ,\delta ,v,c}h1;\tilde{p},\end{array}$$where $\overline{w}=w/{{\varsigma }^p\upsilon -{\varsigma }^p\epsilon }^{\mu }$ and $\theta t={\varsigma }^{1/p}t$ for $t\in {\epsilon }^p,{\upsilon }^p$

(ii) If $p<0$, in that case,

$$\begin{array}{}\text{(15)}& \sigma {\frac{{\varsigma }^p\epsilon +{\varsigma }^p\upsilon }{2}}^{1/p}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\epsilon -}^{\gamma ,\delta ,v,c}1{\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}\le h\frac{1}{2}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\upsilon +}^{\gamma ,\delta ,v,c}\sigma \circ \varsigma {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}+{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\epsilon -}^{\gamma ,\delta ,v,c}\sigma \circ \varsigma {\varsigma }^{-1}{\varsigma }^p\epsilon ;\tilde{p}\le h\frac{1}{2}\sigma \varsigma \epsilon +\sigma \varsigma \upsilon {}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},0-}^{\gamma ,\delta ,v,c}h0;\tilde{p}+{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},1+}^{\gamma ,\delta ,v,c}h1;\tilde{p},\end{array}$$where $\overline{w}=w/{{\varsigma }^p\epsilon -{\varsigma }^p\upsilon }^{\mu }$ and $\theta t={\varsigma }^{1/p}t$ for $t\in {\upsilon }^p,{\epsilon }^p$

Proof.

We demonstrate claim (i) as noted below:

(i) Since $\sigma$ is $p,h$-convex function on $\epsilon ,\upsilon$, for $\zeta ,\eta \in I$, we get

$$\begin{array}{}\text{(16)}& \sigma {\frac{{\varsigma }^p\zeta +{\varsigma }^p\eta }{2}}^{1/p}\le h\frac{1}{2}\sigma \varsigma \zeta +\sigma \varsigma \eta .\end{array}$$

Taking $\varsigma \zeta ={t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}$ and $\varsigma \eta ={t{\varsigma }^p\upsilon +1-t{\varsigma }^p\epsilon }^{1/p}$ in the above inequality, we get $$\begin{array}{}\text{(17)}& \sigma {\frac{{\varsigma }^p\epsilon +{\varsigma }^p\upsilon }{2}}^{1/p}\le h\frac{1}{2}\sigma {t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}+\sigma {t{\varsigma }^p\upsilon +1-t{\varsigma }^p\epsilon }^{1/p}.\end{array}$$

Multiplying both sides of (17) by $t^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}$ and integrating over $0,1$, we get $$\begin{array}{}\text{(18)}& \sigma {\frac{{\varsigma }^p\epsilon +{\varsigma }^p\upsilon }{2}}^{1/p}{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}dt\le h\frac{1}{2}{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}\sigma {t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}dt+h\frac{1}{2}{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}\sigma {t{\varsigma }^p\upsilon +1-t{\varsigma }^p\epsilon }^{1/p}dt.\end{array}$$

By choosing $\varsigma \zeta =t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon$ and $\varsigma \eta =t{\varsigma }^p\upsilon +1-t{\varsigma }^p\epsilon$ in (18), we get $$\begin{array}{}\text{(19)}& \sigma {\frac{{\varsigma }^p\epsilon +{\varsigma }^p\upsilon }{2}}^{1/p}{\int }_{{\varsigma }^{-1}{\varsigma }^p\epsilon }^{{\varsigma }^{-1}{\varsigma }^p\upsilon }{{\varsigma }^p\upsilon -\varsigma \zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{{\varsigma }^p\upsilon -\varsigma \zeta }^{\mu };\tilde{p}{\varsigma }^{\prime }\zeta d\zeta \le h\frac{1}{2}{\int }_{{\varsigma }^{-1}{\varsigma }^p\epsilon }^{{\varsigma }^{-1}{\varsigma }^p\upsilon }{{\varsigma }^p\upsilon -\varsigma \zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{{\varsigma }^p\upsilon -\varsigma \zeta }^{\mu };\tilde{p}\sigma {\varsigma }^{1/p}\zeta {\varsigma }^{\prime }\zeta d\zeta +h\frac{1}{2}{\int }_{{\varsigma }^{-1}{\varsigma }^p\epsilon }^{{\varsigma }^{-1}{\varsigma }^p\upsilon }{\varsigma \eta -{\varsigma }^p\epsilon }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{\varsigma \eta -{\varsigma }^p\epsilon }^{\mu };\tilde{p}\sigma {\varsigma }^{1/p}\eta {\varsigma }^{\prime }\eta d\eta .\end{array}$$

By utilizing $k$-fractional operators (6) and (7), the first side of (14) is acquired.

Now, to demonstrate the second side of (14), once again, $p,h$-convexity of $f$ over $\epsilon ,\upsilon$, and for $t\in 0,1$, we get $$\begin{array}{}\text{(20)}& \sigma {t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}+\sigma {t{\varsigma }^p\upsilon +1-t{\varsigma }^p\epsilon }^{1/p}\le ht+h1-t\sigma \varsigma \epsilon +\sigma \varsigma \upsilon .\end{array}$$

Multiplying both sides of (20) by $h1/2t^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}$ and integrating over $0,1$, we get $$\begin{array}{}\text{(21)}& h\frac{1}{2}{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}\sigma {t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}\sigma {t{\varsigma }^p\upsilon +1-t{\varsigma }^p\epsilon }^{1/p}dt\le h\frac{1}{2}\sigma \varsigma \epsilon +\sigma \varsigma \upsilon \times {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}htdt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}h1-tdt.\end{array}$$

Taking $\varsigma \zeta =t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon$ and $\varsigma \eta =t{\varsigma }^p\upsilon +1-t{\varsigma }^p\epsilon$ in (21), in that case, by utilizing $k$-fractional operators (6) and (7), the second side of (14) is acquired.

(ii) Proof is the same as the proof of (i)

Corollary 13.

By utilizing (14) and (15), some more $k$-fractional inequalities are offered as noted below:

(i) By choosing $\varsigma =I$ and $\tilde{p}=w=0$, we acquire

$$\begin{array}{}\text{(22)}& \sigma {\frac{{\epsilon }^p+{\upsilon }^p}{2}}^{1/p}{\int }_{{\epsilon }^p}^{{\upsilon }^p}{{\upsilon }^p-\zeta }^{\alpha /k-1}d\zeta \le h\frac{1}{2}{\int }_{{\epsilon }^p}^{{\upsilon }^p}{{\upsilon }^p-\zeta }^{\alpha /k-1}\sigma {\zeta }^{1/p}d\zeta +{\int }_{{\epsilon }^p}^{{\upsilon }^p}{\eta -{\epsilon }^p}^{\alpha /k-1}\sigma {\eta }^{1/p}d\eta \le h\frac{1}{2}\sigma \epsilon +\sigma \upsilon {\int }_0^1{t}^{\alpha /k-1}htdt+{\int }_0^1{t}^{\alpha /k-1}h1-tdt\end{array}$$

(ii) By choosing $\varsigma =I$, $\tilde{p}=0,p=-1$, we acquire

$$\begin{array}{}\text{(23)}& \sigma \frac{2\epsilon \upsilon }{\epsilon +\upsilon }{\int }_{1/\epsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{\frac{1}{\upsilon }-\zeta }^{\mu }d\zeta \le h\frac{1}{2}{\int }_{1/\epsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{\frac{1}{\upsilon }-\zeta }^{\mu }\sigma \frac{1}{\zeta }d\zeta +{\int }_{1/\epsilon }^{1/\upsilon }{\eta -\frac{1}{\epsilon }}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{\eta -\frac{1}{\epsilon }}^{\mu }\sigma \frac{1}{\eta }d\eta \le h\frac{1}{2}\sigma \epsilon +\sigma \upsilon {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu }htdt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu }h1-tdt\end{array}$$

(iii) By choosing $p=-1$ and $\varsigma =I$, we acquire

$$\begin{array}{}\text{(24)}& \sigma \frac{2\epsilon \upsilon }{\epsilon +\upsilon }{\int }_{1/\epsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{\frac{1}{\upsilon }-\zeta }^{\mu };\tilde{p}d\zeta \le h\frac{1}{2}{\int }_{1/\epsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{\frac{1}{\upsilon }-\zeta }^{\mu };\tilde{p}\sigma \frac{1}{\zeta }d\zeta +{\int }_{1/\epsilon }^{1/\upsilon }{\eta -\frac{1}{\epsilon }}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{\eta -\frac{1}{\epsilon }}^{\mu };\tilde{p}\sigma \frac{1}{\eta }d\eta \le h\frac{1}{2}\sigma \epsilon +\sigma \upsilon {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}htdt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}h1-tdt\end{array}$$

(iv) By choosing $p=-1$, $w=\tilde{p}=0$, and $\varsigma =I$, we acquire

$$\begin{array}{}\text{(25)}& \sigma \frac{2\epsilon \upsilon }{\epsilon +\upsilon }{\int }_{1/\epsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}d\zeta \le h\frac{1}{2}{\int }_{1/\epsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}\sigma \frac{1}{\zeta }d\zeta +{\int }_{1/\epsilon }^{1/\upsilon }{\eta -\frac{1}{\epsilon }}^{\alpha /k-1}\sigma \frac{1}{\eta }d\eta \le h\frac{1}{2}\sigma \epsilon +\sigma \upsilon {\int }_0^1{t}^{\alpha /k-1}htdt+{\int }_0^1{t}^{\alpha /k-1}h1-tdt\end{array}$$

(v) By choosing $p=-1$, we acquire

$$\begin{array}{}\text{(26)}& \sigma {\frac{{\varsigma }^{-1}\epsilon +{\varsigma }^{-1}\upsilon }{2}}^{-1}\times {\int }_{{\varsigma }^{-1}{\varsigma }^{-1}\epsilon }^{{\varsigma }^{-1}{\varsigma }^{-1}\upsilon }{{\varsigma }^{-1}\upsilon -\varsigma \zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{{\varsigma }^{-1}\upsilon -\varsigma \zeta }^{\mu };\tilde{p}{\varsigma }^{\prime }\zeta d\zeta \le h\frac{1}{2}{\int }_{{\varsigma }^{-1}{\varsigma }^{-1}\epsilon }^{{\varsigma }^{-1}{\varsigma }^{-1}\upsilon }{{\varsigma }^{-1}\upsilon -\varsigma \zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{{\varsigma }^{-1}\upsilon -\varsigma \zeta }^{\mu };\tilde{p}\sigma {\varsigma }^{-1}\zeta {\varsigma }^{\prime }\zeta d\zeta +{\int }_{{\varsigma }^{-1}{\varsigma }^{-1}\epsilon }^{{\varsigma }^{-1}{\varsigma }^{-1}\upsilon }{\varsigma \eta -{\varsigma }^{-1}\epsilon }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{\varsigma \eta -{\varsigma }^{-1}\epsilon }^{\mu };\tilde{p}\sigma {\varsigma }^{-1}\eta {\varsigma }^{\prime \eta }d\eta \le h\frac{1}{2}\sigma \epsilon +\sigma \upsilon {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}htdt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}h1-tdt\end{array}$$

Remark 14.

The well-known $k$-fractional inequalities are further noted as below:

(i) By choosing $ht=t$ and $k=1$ in Corollary 13 (i), Theorem 9 of [6] is acquired

(ii) By choosing $ht=t$ and $k=1$ in Corollary 13 (ii), Theorem 2.1 of [24] is acquired

(iii) By choosing $ht=t$ and $k=1$ in Corollary 13 (iii), Theorem 2.1 of [5] is acquired

(iv) By choosing $ht=t$ and $k=1$ in Corollary 13 (iv), Theorem 4 of [1] is acquired

(v) By choosing $ht=t$ and $k=1$ in Corollary 13 (v), Theorem 2.1 of [27] is acquired

The below lemma is beneficial to offer the Fejér-Hadamard inequality for generalized $k$-fractional integrals.

Lemma 15 (see [18]).

Let $\sigma ,\varsigma :\epsilon ,\upsilon ⟶ℝ$ with $0<\epsilon <\upsilon$ be the functions such that $\sigma$ is positive and $\sigma \in L_1\epsilon ,\upsilon$, and $\varsigma$ is differentiable and absolutely increasing. If $p\in R\backslash 0$ and $\sigma {\varsigma }^{1/p}\zeta =\sigma {{\varsigma }^p\epsilon +{\varsigma }^p\upsilon -\varsigma \zeta }^{1/p}$, in the case for generalized $k$-fractional operators (6) and (7), we have

(i) If $p>0$, in that case,

$$\begin{array}{}\text{(27)}& {}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\epsilon +}^{\gamma ,\delta ,v,c}\sigma \circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}={}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\upsilon -}^{\gamma ,\delta ,v,c}\sigma \circ \theta {\varsigma }^{-1}{\varsigma }^p\epsilon ;\tilde{p}=\frac{1}{2}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\epsilon +}^{\gamma ,\delta ,v,c}\sigma \circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}+{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\upsilon -}^{\gamma ,\delta ,v,c}\sigma \circ \theta {\varsigma }^{-1}{\varsigma }^p\epsilon ;\tilde{p},\end{array}$$with $\theta t={\varsigma }^{1/p}t$ for all $t\in {\epsilon }^p,{\upsilon }^p$

(ii) If $p<0$, in that case,

$$\begin{array}{}\text{(28)}& {}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\upsilon +}^{\gamma ,\delta ,v,c}\sigma \circ \theta {\varsigma }^{-1}{\varsigma }^p\epsilon ;\tilde{p}={}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\epsilon -}^{\gamma ,\delta ,v,c}\sigma \circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}=\frac{1}{2}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\upsilon +}^{\gamma ,\delta ,v,c}\sigma \circ \theta {\varsigma }^{-1}{\varsigma }^p\epsilon ;\tilde{p}+{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\epsilon -}^{\gamma ,\delta ,v,c}\sigma \circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p},\end{array}$$with $\theta t={\varsigma }^{1/p}t$ for all $t\in {\upsilon }^p,{\epsilon }^p$

The Fejér-Hadamard inequality for generalized $k$-fractional integrals is stated and proved in the following theorem.

Theorem 16.

Let $h:J⟶ℝ$ be a nonnegative and nonzero function. Also, let $\sigma ,\varsigma ,\text{r}:\epsilon ,\upsilon ⟶ℝ$, $0<\epsilon <\upsilon$, be the functions such that $\sigma$ is positive and $\sigma \in L_1\epsilon ,\upsilon$, $\varsigma$ is differentiable and absolutely increasing, and $r$ is a nonnegative and integrable function. If $\sigma$ is $p,h$-convex with $p\in R\backslash 0$, and $\sigma {\varsigma }^{1/p}\zeta =\sigma {{\varsigma }^p\epsilon +{\varsigma }^p\upsilon -\varsigma \zeta }^{1/p}$, then the following inequalities for generalized $k$-fractional integral operators (6) and (7) hold:

(i) If $p>0$, in that case,

$$\begin{array}{}\text{(29)}& \sigma {\frac{{\varsigma }^p\epsilon +{\varsigma }^p\upsilon }{2}}^{1/p}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\epsilon +}^{\gamma ,\delta ,v,c}r\circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}+{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\upsilon -}^{\gamma ,\delta ,v,c}r\circ \theta {\varsigma }^{-1}{\varsigma }^p\epsilon ;\tilde{p}\le h\frac{1}{2}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\epsilon +}^{\gamma ,\delta ,v,c}\sigma \circ r\circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}+{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\upsilon -}^{\gamma ,\delta ,v,c}\sigma \circ r\circ \theta {\varsigma }^{-1}{\varsigma }^p\epsilon ;\tilde{p}\le h\frac{1}{2}\sigma \varsigma \epsilon +\sigma \varsigma \upsilon \times {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}htdt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}h1-tdt,\end{array}$$where $\overline{w}=w/{{\varsigma }^p\upsilon -{\varsigma }^p\epsilon }^{\mu }$, and $\theta t={\varsigma }^{1/p}t$ for $t\in {\epsilon }^p,{\upsilon }^p$

(ii) If $p<0$, in that case,

$$\begin{array}{}\text{(30)}& \sigma {\frac{{\varsigma }^p\epsilon +{\varsigma }^p\upsilon }{2}}^{1/p}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\upsilon +}^{\gamma ,\delta ,v,c}r\circ \theta {\varsigma }^{-1}{\varsigma }^p\epsilon ;\tilde{p}+{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\epsilon -}^{\gamma ,\delta ,v,c}r\circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}\le h\frac{1}{2}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\upsilon +}^{\gamma ,\delta ,v,c}\sigma \circ r\circ \theta {\varsigma }^{-1}{\varsigma }^p\epsilon ;\tilde{p}+{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\epsilon -}^{\gamma ,\delta ,v,c}\sigma \circ r\circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}\le h\frac{1}{2}\sigma \varsigma \epsilon +\sigma \varsigma \upsilon {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}htdt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}h1-tdt,\end{array}$$where $\overline{w}=w/{{\varsigma }^p\epsilon -{\varsigma }^p\upsilon }^{\mu }$, and $\theta t={\varsigma }^{1/p}t$ for $t\in {\upsilon }^p,{\epsilon }^p$

Proof.

We demonstrate claim (i) as noted below:

(i) Multiplying both sides of (17) by $t^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}$ and then integrating over $0,1$, we have

$$\begin{array}{}\text{(31)}& \sigma {\frac{{\varsigma }^p\epsilon +{\varsigma }^p\upsilon }{2}}^{1/p}{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}dt\le h\frac{1}{2}{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}\sigma {t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}r{t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}\sigma {t{\varsigma }^p\upsilon +1-t{\varsigma }^p\epsilon }^{1/p}r{t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}dt.\end{array}$$

By choosing $\varsigma x=t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon$, that is, ${\varsigma }^p\epsilon +{\varsigma }^p\upsilon -\varsigma \zeta =t{\varsigma }^p\upsilon +1-t{\varsigma }^p\epsilon$, in (31) and using $\sigma {\varsigma }^{1/p}\zeta =\sigma {{\varsigma }^p\epsilon +{\varsigma }^p\upsilon -\varsigma \zeta }^{1/p}$, we have $$\begin{array}{}\text{(32)}& \sigma {\frac{{\varsigma }^p\epsilon +{\varsigma }^p\upsilon }{2}}^{1/p}\times {\int }_{{\varsigma }^{-1}{\varsigma }^p\epsilon }^{{\varsigma }^{-1}{\varsigma }^p\upsilon }{{\varsigma }^p\upsilon -\varsigma \zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{{\varsigma }^p\upsilon -\varsigma \zeta }^{\mu };\tilde{p}r\circ \theta \zeta {\varsigma }^{\prime }\zeta d\zeta \le h\frac{1}{2}{\int }_{{\varsigma }^{-1}{\varsigma }^p\epsilon }^{{\varsigma }^{-1}{\varsigma }^p\upsilon }{{\varsigma }^p\upsilon -\varsigma \zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{{\varsigma }^p\upsilon -\varsigma \zeta }^{\mu };\tilde{p}\sigma \circ \theta \zeta r\circ \theta \zeta {\varsigma }^{\prime \zeta }d\zeta +{\int }_{{\varsigma }^{-1}{\varsigma }^p\epsilon }^{{\varsigma }^{-1}{\varsigma }^p\upsilon }{\varsigma \zeta -{\varsigma }^p\epsilon }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{\varsigma \zeta -{\varsigma }^p\epsilon }^{\mu };\tilde{p}\sigma \circ \theta \zeta r\circ \theta \zeta {\varsigma }^{\prime \zeta }d\zeta .\end{array}$$

This implies $$\begin{array}{}\text{(33)}& \sigma {\frac{{\varsigma }^p\epsilon +{\varsigma }^p\upsilon }{2}}^{1/p}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\epsilon +}^{\gamma ,\delta ,v,c}r\circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}\le h\frac{1}{2}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\epsilon +}^{\gamma ,\delta ,v,c}\sigma \circ r\circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}+{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\upsilon -}^{\gamma ,\delta ,v,c}\sigma \circ r\circ \theta {\varsigma }^{-1}{\varsigma }^p\epsilon ;\tilde{p}.\end{array}$$

By using Lemma 15 (i) in the above inequality, we get the first inequality of (29).

Now, to demonstrate the second side of (29), multiplying both sides of (20) with $t^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}$, and next integrating over $0,1$, we obtain $$\begin{array}{}\text{(34)}& {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}\sigma {t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}\sigma {t{\varsigma }^p\upsilon +1-t{\varsigma }^p\epsilon }^{1/p}dt\le h\frac{1}{2}\sigma \varsigma \epsilon +\sigma \varsigma \upsilon {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}ht+h1-tdt.\end{array}$$

Setting $\varsigma \zeta =t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon$ and using $\sigma {\varsigma }^{1/p}\zeta =\sigma {{\varsigma }^p\epsilon +{\varsigma }^p\upsilon -\varsigma \zeta }^{1/p}$ in (34), we have $$\begin{array}{}\text{(35)}& {}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\epsilon +}^{\gamma ,\delta ,v,c}\sigma \circ r\circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}+{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{w},{\varsigma }^{-1}{\varsigma }^p\upsilon -}^{\gamma ,\delta ,v,c}\sigma \circ r\circ \theta {\varsigma }^{-1}{\varsigma }^p\epsilon ;\tilde{p}\le h\frac{1}{2}\sigma \varsigma \epsilon +\sigma \varsigma \upsilon \times {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{t{\varsigma }^p\epsilon +1-t{\varsigma }^p\upsilon }^{1/p}ht+h1-tdt.\end{array}$$

By utilizing Lemma 15 (i) in the above inequality, we get the second side of (29).

(ii) The proof is similar to the proof of (i)

Corollary 17.

By utilizing (29) and (30), some more $k$-fractional inequalities are offered as noted below:

(i) By choosing $\varsigma =I$ and $\tilde{p}=0$, we acquire

$$\begin{array}{}\text{(36)}& \sigma {\frac{{\epsilon }^p+{\upsilon }^p}{2}}^{1/p}{\int }_{{\epsilon }^p}^{{\upsilon }^p}{{\upsilon }^p-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{{\upsilon }^p-\zeta }^{\mu }r\circ \theta \zeta d\zeta \le h\frac{1}{2}{\int }_{{\epsilon }^p}^{{\upsilon }^p}{{\upsilon }^p-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{{\upsilon }^p-\zeta }^{\mu }\sigma \circ \theta \zeta r\circ \theta \zeta d\zeta +{\int }_{{\epsilon }^p}^{{\upsilon }^p}{\zeta -{\epsilon }^p}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{\zeta -{\epsilon }^p}^{\mu }\sigma \circ \theta \zeta r\circ \theta \zeta d\zeta \le h\frac{1}{2}\sigma \epsilon +\sigma \upsilon {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu }r{t{\epsilon }^p+1-t{\upsilon }^p}^{1/p}htdt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu }r{t{\epsilon }^p+1-t{\upsilon }^p}^{1/p}h1-tdt\end{array}$$

(ii) By choosing $\varsigma =I$ and $w=\tilde{p}=0$, we acquire

$$\begin{array}{}\text{(37)}& \sigma {\frac{{\epsilon }^p+{\upsilon }^p}{2}}^{1/p}{\int }_{{\epsilon }^p}^{{\upsilon }^p}{{\upsilon }^p-\zeta }^{\alpha /k-1}r\circ \theta \zeta d\zeta \le h\frac{1}{2}{\int }_{{\epsilon }^p}^{{\upsilon }^p}{{\upsilon }^p-\zeta }^{\alpha /k-1}\sigma \circ \theta \zeta r\circ \theta \zeta d\zeta +{\int }_{{\epsilon }^p}^{{\upsilon }^p}{\zeta -{\epsilon }^p}^{\alpha /k-1}\sigma \circ \theta \zeta r\circ \theta \zeta d\zeta \le h\frac{1}{2}\sigma \epsilon +\sigma \upsilon {\int }_0^1{t}^{\alpha /k-1}r{t{\epsilon }^p+1-t{\upsilon }^p}^{1/p}htdt+{\int }_0^1{t}^{\alpha /k-1}r{t{\epsilon }^p+1-t{\upsilon }^p}^{1/p}h1-tdt\end{array}$$

(iii) By choosing $\tilde{p}=0$, $r\zeta =1$, $\varsigma =I$, and $p=-1$, we acquire

$$\begin{array}{}\text{(38)}& \sigma \frac{2\epsilon \upsilon }{\epsilon +\upsilon }{\int }_{1/\epsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{\frac{1}{\upsilon }-\zeta }^{\mu }\theta \zeta d\zeta \le h\frac{1}{2}{\int }_{1/\epsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{\frac{1}{\upsilon }-\zeta }^{\mu }\sigma \circ \theta \zeta \theta \zeta d\zeta +{\int }_{1/\epsilon }^{1/\upsilon }{\zeta -\frac{1}{\epsilon }}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{\zeta -\frac{1}{\epsilon }}^{\mu }\sigma \circ \theta \zeta \theta \zeta d\zeta \le h\frac{1}{2}\sigma \epsilon +\sigma \upsilon {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu }htdt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu }h1-tdt\end{array}$$

(iv) By choosing $\varsigma =I$, $p=-1$, and $r\zeta =1$, we acquire

$$\begin{array}{}\text{(39)}& \sigma \frac{2\epsilon \upsilon }{\epsilon +\upsilon }{\int }_{1/\epsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{\frac{1}{\upsilon }-\zeta }^{\mu };\tilde{p}\theta \zeta d\zeta \le h\frac{1}{2}{\int }_{1/\epsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{\frac{1}{\upsilon }-\zeta }^{\mu };\tilde{p}\sigma \circ \theta \zeta \theta \zeta d\zeta +{\int }_{1/\epsilon }^{1/\upsilon }{\zeta -\frac{1}{\epsilon }}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{\zeta -\frac{1}{\epsilon }}^{\mu };\tilde{p}\sigma \circ \theta \zeta \theta \zeta d\zeta \le h\frac{1}{2}\sigma \epsilon +\sigma \upsilon {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}htdt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}h1-tdt\end{array}$$

(v) By choosing $w=\tilde{p}=0$, $r\zeta =1$, $p=-1$, and $\varsigma =I$, we acquire

$$\begin{array}{}\text{(40)}& \sigma \frac{2\epsilon \upsilon }{\epsilon +\upsilon }{\int }_{1/\epsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}\theta \zeta d\zeta \le h\frac{1}{2}{\int }_{1/\epsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}\sigma \circ \theta \zeta \theta \zeta d\zeta +{\int }_{1/\epsilon }^{1/\upsilon }{\zeta -\frac{1}{\epsilon }}^{\alpha /k-1}\sigma \circ \theta \zeta \theta \zeta d\zeta \le h\frac{1}{2}\sigma \epsilon +\sigma \upsilon {\int }_0^1{t}^{\alpha /k-1}htdt+{\int }_0^1{t}^{\alpha /k-1}h1-tdt\end{array}$$

(vi) By choosing $p=-1$, we acquire

$$\begin{array}{}\text{(41)}& \sigma {\frac{{\varsigma }^{-1}\epsilon +{\varsigma }^{-1}\upsilon }{2}}^{-1}\times {\int }_{{\varsigma }^{-1}{\varsigma }^{-1}\epsilon }^{{\varsigma }^{-1}{\varsigma }^{-1}\upsilon }{{\varsigma }^{-1}\upsilon -\varsigma \zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{{\varsigma }^{-1}\upsilon -\varsigma \zeta }^{\mu };\tilde{p}r\circ \theta \zeta {\varsigma }^{\prime \zeta }d\zeta \le h\frac{1}{2}{\int }_{{\varsigma }^{-1}{\varsigma }^{-1}\epsilon }^{{\varsigma }^{-1}{\varsigma }^{-1}\upsilon }{{\varsigma }^{-1}\upsilon -\varsigma \zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{{\varsigma }^{-1}\upsilon -\varsigma \zeta }^{\mu };\tilde{p}\sigma \circ \theta \zeta r\circ \theta \zeta {\varsigma }^{\prime \zeta }d\zeta +{\int }_{{\varsigma }^{-1}{\varsigma }^{-1}\epsilon }^{{\varsigma }^{-1}{\varsigma }^{-1}\upsilon }{\varsigma \zeta -{\varsigma }^{-1}\epsilon }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{w}{\varsigma \zeta -{\varsigma }^{-1}\epsilon }^{\mu };\tilde{p}\sigma \circ \theta \zeta r\circ \theta \zeta {\varsigma }^{\prime \zeta }d\zeta \le h\frac{1}{2}\sigma \varsigma \epsilon +\sigma \varsigma \upsilon {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r\frac{\varsigma \epsilon \varsigma \upsilon }{t\varsigma \upsilon +1-t\varsigma \epsilon }htdt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r\frac{\varsigma \epsilon \varsigma \upsilon }{t\varsigma \upsilon +1-t\varsigma \epsilon }h1-tdt\end{array}$$

Remark 18.

The mentioned $k$-fractional inequalities are further connected with foreknown conclusions as noted below:

(i) By setting $ht=t$ and $k=1$ in Corollary 17 (ii), Theorem 9 of [6] is obtained

(ii) By setting $ht=t$ and $k=1$ in Corollary 17 (iii), Theorem 2.1 of [24] is obtained

(iii) By setting $ht=t$ and $k=1$ in Corollary 17 (iv), Theorem 2.1 of [5] is obtained

(iv) By setting $ht=t$ and $k=1$ in Corollary 17 (v), Theorem 4 of [1] is obtained

(v) By setting $ht=t$ and $k=1$ in Corollary 17 (vi), Theorem 2.5 of [27] is obtained

In the next theorem, we offer another type of the Hadamard inequality.

Theorem 19.

Let $h:J⟶ℝ$ is a nonnegative and nonzero function. Also, let $\sigma ,\varsigma :\epsilon ,\upsilon ⟶ℝ$, $0<\epsilon <\upsilon$, be the functions such that $\sigma$ is positive and $\sigma \in L_1\epsilon ,\upsilon$ and $\varsigma$ is differentiable and absolutely increasing. If $\sigma$ is $p,h$-convex, $p\in R\backslash 0$, then for generalized $k$-fractional integral operators (6) and (7), we have the following:

(i) If $p>0$, in that case,

$$\begin{array}{}\text{(42)}& \sigma {\frac{{\varsigma }^p\epsilon +{\varsigma }^p\upsilon }{2}}^{1/p}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{\overline{w}},{\varsigma }^{-1}{\varsigma }^p\epsilon +{\varsigma }^p\upsilon /2+}^{\gamma ,\delta ,v,c}1{\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}\le h\frac{1}{2}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{\overline{w}},{\varsigma }^{-1}{\varsigma }^p\epsilon +{\varsigma }^p\upsilon /2+}^{\gamma ,\delta ,v,c}\sigma \circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}+{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{\overline{w}},{\varsigma }^{-1}{\varsigma }^p\epsilon +{\varsigma }^p\upsilon /2-}^{\gamma ,\delta ,v,c}\sigma \circ \theta {\varsigma }^{-1}{\varsigma }^p\epsilon ;\tilde{p}\le h\frac{1}{2}\sigma \varsigma \epsilon +\sigma \varsigma \upsilon \times {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}h\frac{t}{2}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}h\frac{2-t}{2}dt,\end{array}$$where $\overline{\overline{w}}=2^{\mu }w/{{\varsigma }^p\upsilon -{\varsigma }^p\epsilon }^{\mu }$, and $\theta t={\varsigma }^{1/p}t$ for $t\in {\epsilon }^p,{\upsilon }^p$

(ii) If $p<0$, in that case,

$$\begin{array}{}\text{(43)}& \sigma {\frac{{\varsigma }^p\epsilon +{\varsigma }^p\upsilon }{2}}^{1/p}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{\overline{w}},{\varsigma }^{-1}{\varsigma }^p\epsilon +{\varsigma }^p\upsilon /2-}^{\gamma ,\delta ,v,c}1{\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}\le h\frac{1}{2}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{\overline{w}},{\varsigma }^{-1}{\varsigma }^p\epsilon +{\varsigma }^p\upsilon /2+}^{\gamma ,\delta ,v,c}\sigma \circ \theta {\varsigma }^{-1}{\varsigma }^p\epsilon ;\tilde{p}+{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{\overline{w}},{\varsigma }^{-1}{\varsigma }^p\epsilon +{\varsigma }^p\upsilon /2-}^{\gamma ,\delta ,v,c}\sigma \circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}\le h\frac{1}{2}\sigma \varsigma \epsilon +\sigma \varsigma \upsilon \times {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}h\frac{t}{2}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}h\frac{2-t}{2}dt,\end{array}$$where $\overline{\overline{w}}=2^{\mu }w/{{\varsigma }^p\epsilon -{\varsigma }^p\upsilon }^{\mu }$, and $\theta t={\varsigma }^{1/p}t$ for $t\in {\upsilon }^p,{\epsilon }^p$

Proof.

We demonstrate claim (i) as noted below:

(i) Choosing $\varsigma \zeta ={t/2{\varsigma }^p\epsilon +2-t/2{\varsigma }^p\upsilon }^{1/p}$ and $\varsigma \eta ={t/2{\varsigma }^p\upsilon +2-t/2{\varsigma }^p\epsilon }^{1/p}$ in (16), we have

$$\begin{array}{}\text{(44)}& \sigma {\frac{{\varsigma }^p\epsilon +{\varsigma }^p\upsilon }{2}}^{1/p}\le h\frac{1}{2}\sigma {\frac{t}{2}{\varsigma }^p\epsilon +\frac{2-t}{2}{\varsigma }^p\upsilon }^{1/p}+\sigma {\frac{t}{2}{\varsigma }^p\upsilon +\frac{2-t}{2}{\varsigma }^p\epsilon }^{1/p}.\end{array}$$

Multiplying both sides of (44) by $t^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}$ and then integrating over $0,1$, we have $$\begin{array}{}\text{(45)}& \sigma {\frac{{\varsigma }^p\epsilon +{\varsigma }^p\upsilon }{2}}^{1/p}{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}dt\le h\frac{1}{2}{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}\sigma {\frac{t}{2}{\varsigma }^p\epsilon +\frac{2-t}{2}{\varsigma }^p\upsilon }^{1/p}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}\sigma {\frac{t}{2}{\varsigma }^p\upsilon +\frac{2-t}{2}{\varsigma }^p\epsilon }^{1/p}dt.\end{array}$$

By choosing $\varsigma \zeta ={t/2{\varsigma }^p\epsilon +2-t/2{\varsigma }^p\upsilon }^{1/p}$ and $\varsigma \eta ={t/2{\varsigma }^p\upsilon +2-t/2{\varsigma }^p\epsilon }^{1/p}$ in (45), then by using $k$-fractional integral operators (6) and (7), the first inequality of (42) is obtained.

Now, to demonstrate the second side of (42), once again $p,h$-convexity of $\sigma$ over $\epsilon ,\upsilon$ and for $t\in 0,1$, we obtain $$\begin{array}{}\text{(46)}& \sigma {\frac{t}{2}{\varsigma }^p\epsilon +\frac{2-t}{2}{\varsigma }^p\upsilon }^{1/p}+\sigma {\frac{t}{2}{\varsigma }^p\upsilon +\frac{2-t}{2}{\varsigma }^p\epsilon }^{1/p}\le \sigma \varsigma \epsilon +\sigma \varsigma \upsilon h\frac{t}{2}+h\frac{2-t}{2}.\end{array}$$

Multiplying both sides of (46) by $h1/2t^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}$ and then integrating over $0,1$, we have $$\begin{array}{}\text{(47)}& h\frac{1}{2}{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}\sigma {\frac{t}{2}{\varsigma }^p\epsilon +\frac{2-t}{2}{\varsigma }^p\upsilon }^{1/p}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}\sigma {\frac{t}{2}{\varsigma }^p\upsilon +\frac{2-t}{2}{\varsigma }^p\epsilon }^{1/p}dt\le h\frac{1}{2}\sigma \varsigma \epsilon +\sigma \varsigma \upsilon \times {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}h\frac{t}{2}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}h\frac{2-t}{2}dt.\end{array}$$

Choosing $\varsigma \zeta =t/2{\varsigma }^p\epsilon +2-t/2{\varsigma }^p\upsilon$ and $\varsigma \eta =t/2{\varsigma }^p\upsilon +2-t/2{\varsigma }^p\epsilon$ in (47) and by utilizing $k$-fractional operators (6) and (7), the second side of (42) is acquired.

(ii) Proof is the same as the proof of (i)

Corollary 20.

By utilizing (42) and (43), some more $k$-fractional inequalities are offered as noted below:

(i) By choosing $\varsigma =I$ and $\tilde{p}=0,p=-1$, we acquire

$$\begin{array}{}\text{(48)}& \sigma \frac{2\epsilon \upsilon }{\epsilon +\upsilon }{\int }_{2\epsilon \upsilon /\epsilon +\upsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{\frac{1}{\upsilon }-\zeta }^{\mu }d\zeta \le h\frac{1}{2}{\int }_{2\epsilon \upsilon /\epsilon +\upsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{\frac{1}{\upsilon }-\zeta }^{\mu }\sigma {\zeta }^{-1}d\zeta +{\int }_{1/\epsilon }^{2\epsilon \upsilon /\epsilon +\upsilon }{\eta -\frac{1}{\epsilon }}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{\eta -\frac{1}{\epsilon }}^{\mu }\sigma {\eta }^{-1}d\eta \le h\frac{1}{2}\sigma \epsilon +\sigma \upsilon \times {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu }h\frac{t}{2}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu }h\frac{2-t}{2}dt\end{array}$$

(ii) By choosing $p=-1$ and $\varsigma =I$, we acquire

$$\begin{array}{}\text{(49)}& \sigma \frac{2\epsilon \upsilon }{\epsilon +\upsilon }{\int }_{2\epsilon \upsilon /\epsilon +\upsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{\frac{1}{\upsilon }-\zeta }^{\mu };\tilde{p}d\zeta \le h\frac{1}{2}{\int }_{2\epsilon \upsilon /\epsilon +\upsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{\frac{1}{\upsilon }-\zeta }^{\mu };\tilde{p}\sigma {\zeta }^{-1}d\zeta +{\int }_{1/\epsilon }^{2\epsilon \upsilon /\epsilon +\upsilon }{\eta -\frac{1}{\epsilon }}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{\eta -\frac{1}{\epsilon }}^{\mu };\tilde{p}\sigma {\eta }^{-1}d\eta \le h\frac{1}{2}\sigma \epsilon +\sigma \upsilon \times {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}h\frac{t}{2}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}h\frac{2-t}{2}dt\end{array}$$

(iii) By choosing $p=-1$, we acquire

$$\begin{array}{}\text{(50)}& \sigma {\frac{{\varsigma }^{-1}\epsilon +{\varsigma }^{-1}\upsilon }{2}}^{-1}\times {\int }_{{\varsigma }^{-1}{\varsigma }^{-1}\epsilon +{\varsigma }^{-1}\upsilon /2}^{{\varsigma }^{-1}{\varsigma }^{-1}\upsilon }{{\varsigma }^{-1}\upsilon -\varsigma \zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{{\varsigma }^{-1}\upsilon -\varsigma \zeta }^{\mu };\tilde{p}{\varsigma }^{\prime \zeta }d\zeta \le h\frac{1}{2}{\int }_{{\varsigma }^{-1}{\varsigma }^{-1}\epsilon +{\varsigma }^{-1}\upsilon /2}^{{\varsigma }^{-1}{\varsigma }^{-1}\upsilon }{{\varsigma }^{-1}\upsilon -\varsigma \zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{{\varsigma }^{-1}\upsilon -\varsigma \zeta }^{\mu };\tilde{p}\sigma {\varsigma }^{-1}\zeta {\varsigma }^{\prime \zeta }d\zeta +{\int }_{{\varsigma }^{-1}{\varsigma }^{-1}\epsilon }^{{\varsigma }^{-1}{\varsigma }^{-1}\epsilon +{\varsigma }^{-1}\upsilon /2}{\varsigma \eta -{\varsigma }^{-1}\epsilon }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{\varsigma \eta -{\varsigma }^{-1}\epsilon }^{\mu };\tilde{p}\sigma {\varsigma }^{-1}\eta {\varsigma }^{\prime \eta }d\eta \le h\frac{1}{2}\sigma \varsigma \epsilon +\sigma \varsigma \upsilon \times {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}h\frac{t}{2}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}h\frac{2-t}{2}dt\end{array}$$

Remark 21.

The mentioned $k$-fractional inequalities are farther connected with foreknown conclusions as noted below:

(i) By choosing $ht=t$ and $k=1$ in Corollary 20 (i), Theorem 2.3 of [24] is acquired

(ii) By choosing $ht=t$ and $k=1$ in Corollary 20 (ii), Theorem 2.3 of [5] is acquired

(iii) By choosing $ht=t$ and $k=1$ in Corollary 20 (iii), Theorem 2.7 of [27] is acquired

The second type of the Fejér-Hadamard inequality for generalized $k$-fractional integrals is given as noted below.

Theorem 22.

Let $h:J⟶ℝ$ be a nonnegative and nonzero function. Also, let $\sigma ,\varsigma ,r:\epsilon ,\upsilon ⟶ℝ$ with $0<\epsilon <\upsilon$ be the functions such that $\sigma$ is positive and $\sigma \in L_1\epsilon ,\upsilon$, $\varsigma$ is differentiable and absolutely increasing, and $r$ is a nonnegative and integrable function. If $\sigma$ is $p,h$-convex, $p\in R\backslash 0$ and $\sigma {\varsigma }^{1/p}\zeta =\sigma {{\varsigma }^p\epsilon +{\varsigma }^p\upsilon -\varsigma \zeta }^{1/p}$, then the following inequalities for generalized $k$-fractional integral operators (6) and (7) hold:

(i) If $p>0$, in that case,

$$\begin{array}{}\text{(51)}& \sigma {\frac{{\varsigma }^p\epsilon +{\varsigma }^p\upsilon }{2}}^{1/p}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{\overline{w}},{\varsigma }^{-1}{\varsigma }^p\epsilon +{\varsigma }^p\upsilon /2+}^{\gamma ,\delta ,v,c}r\circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}\le h\frac{1}{2}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{\overline{w}},{\varsigma }^{-1}{\varsigma }^p\epsilon +{\varsigma }^p\upsilon /2+}^{\gamma ,\delta ,v,c}r\circ \sigma \circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}+{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{\overline{w}},{\varsigma }^{-1}{\varsigma }^p\epsilon +{\varsigma }^p\upsilon /2-}^{\gamma ,\delta ,v,c}r\circ \sigma \circ \theta {\varsigma }^{-1}{\varsigma }^p\epsilon ;\tilde{p}\le h\frac{1}{2}\sigma \varsigma \epsilon +\sigma \varsigma \upsilon \times {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{\frac{t}{2}{\varsigma }^p\epsilon +\frac{2-t}{2}{\varsigma }^p\upsilon }^{1/p}h\frac{t}{2}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{\frac{t}{2}{\varsigma }^p\epsilon +\frac{2-t}{2}{\varsigma }^p\upsilon }^{1/p}h\frac{2-t}{2}dt,\end{array}$$where $\overline{\overline{w}}=2^{\mu }w/{{\varsigma }^p\upsilon -{\varsigma }^p\epsilon }^{\mu }$, and $\theta t={\varsigma }^{1/p}t$ for $t\in {\epsilon }^p,{\upsilon }^p$

(ii) If $p<0$, in that case,

$$\begin{array}{}\text{(52)}& \sigma {\frac{{\varsigma }^p\epsilon +{\varsigma }^p\upsilon }{2}}^{1/p}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{\overline{w}},{\varsigma }^{-1}{\varsigma }^p\epsilon +{\varsigma }^p\upsilon /2-}^{\gamma ,\delta ,v,c}r\circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}\le h\frac{1}{2}{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{\overline{w}},{\varsigma }^{-1}{\varsigma }^p\epsilon +{\varsigma }^p\upsilon /2+}^{\gamma ,\delta ,v,c}r\circ f\circ \theta {\varsigma }^{-1}{\varsigma }^p\epsilon ;\tilde{p}+{}_{\varsigma }^k{F}_{\mu ,\alpha ,l,\overline{\overline{w}},{\varsigma }^{-1}{\varsigma }^p\epsilon +{\varsigma }^p\upsilon /2-}^{\gamma ,\delta ,v,c}r\circ \sigma \circ \theta {\varsigma }^{-1}{\varsigma }^p\upsilon ;\tilde{p}\le h\frac{1}{2}\sigma \varsigma \epsilon +\sigma \varsigma \upsilon \times {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{\frac{t}{2}{\varsigma }^p\epsilon +\frac{2-t}{2}{\varsigma }^p\upsilon }^{1/p}h\frac{t}{2}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{\frac{t}{2}{\varsigma }^p\epsilon +\frac{2-t}{2}{\varsigma }^p\upsilon }^{1/p}h\frac{2-t}{2}dt,\end{array}$$where $\overline{\overline{w}}=2^{\mu }w/{{\varsigma }^p\epsilon -{\varsigma }^p\upsilon }^{\mu }$, and $\theta t={\varsigma }^{1/p}t$ for $t\in {\upsilon }^p,{\epsilon }^p$

Proof.

We demonstrate the first claim as noted below:

(i) Multiplying (44) by $t^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{t/2{\varsigma }^p\epsilon +2-t/2{\varsigma }^p\upsilon }^{1/p}$ and then integrating over $0,1$, we have

$$\begin{array}{}\text{(53)}& \sigma {\frac{{\varsigma }^p\epsilon +{\varsigma }^p\upsilon }{2}}^{1/p}{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{\frac{t}{2}{\varsigma }^p\epsilon +\frac{2-t}{2}{\varsigma }^p\upsilon }^{1/p}dt\le h\frac{1}{2}{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{\frac{t}{2}{\varsigma }^p\epsilon +\frac{2-t}{2}{\varsigma }^p\upsilon }^{1/p}\sigma {\frac{t}{2}{\varsigma }^p\epsilon +\frac{2-t}{2}{\varsigma }^p\upsilon }^{1/p}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{\frac{t}{2}{\varsigma }^p\epsilon +\frac{2-t}{2}{\varsigma }^p\upsilon }^{1/p}\sigma {\frac{t}{2}{\varsigma }^p\upsilon +\frac{2-t}{2}{\varsigma }^p\epsilon }^{1/p}dt.\end{array}$$

By choosing $\varsigma \zeta =t/2{\varsigma }^p\epsilon +2-t/2{\varsigma }^p\upsilon$ and $\varsigma \eta =t/2{\varsigma }^p\upsilon +2-t/2{\varsigma }^p\epsilon$, that is, ${\varsigma }^p\epsilon +{\varsigma }^p\upsilon -\varsigma \zeta =t/2{\varsigma }^p\upsilon +2-t/2{\varsigma }^p\epsilon$, in (53), in that case, by utilizing $\sigma {\varsigma }^{1/p}\zeta =\sigma {{\varsigma }^p\epsilon +{\varsigma }^p\upsilon -\varsigma \zeta }^{1/p}$ and $k$-fractional integral operators (6) and (7), the first side of (51) is obtained.

Now, to demonstrate the second side of (51), multiplying both sides of (46) with $h1/2t^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{t/2{\varsigma }^p\epsilon +2-t/2{\varsigma }^p\upsilon }^{1/p}$ and integrating over $0,1$, we obtain $$\begin{array}{}\text{(54)}& h\frac{1}{2}{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{\frac{t}{2}{\varsigma }^p\epsilon +\frac{2-t}{2}{\varsigma }^p\upsilon }^{1/p}\sigma {\frac{t}{2}{\varsigma }^p\epsilon +\frac{2-t}{2}{\varsigma }^p\upsilon }^{1/p}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{\frac{t}{2}{\varsigma }^p\epsilon +\frac{2-t}{2}{\varsigma }^p\upsilon }^{1/p}\sigma {\frac{t}{2}{\varsigma }^p\upsilon +\frac{2-t}{2}{\varsigma }^p\epsilon }^{1/p}dt\le h\frac{1}{2}\sigma \varsigma \epsilon +\sigma \varsigma \upsilon \times {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{\frac{t}{2}{\varsigma }^p\epsilon +\frac{2-t}{2}{\varsigma }^p\upsilon }^{1/p}h\frac{t}{2}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r{\frac{t}{2}{\varsigma }^p\epsilon +\frac{2-t}{2}{\varsigma }^p\upsilon }^{1/p}h\frac{2-t}{2}dt.\end{array}$$

By choosing $\varsigma \zeta =t/2{\varsigma }^p\epsilon +2-t/2{\varsigma }^p\upsilon$ and $\varsigma \eta =t/2{\varsigma }^p\upsilon +2-t/2{\varsigma }^p\epsilon$, that is, ${\varsigma }^p\epsilon +{\varsigma }^p\upsilon -\varsigma \zeta =t/2{\varsigma }^p\upsilon +2-t/2{\varsigma }^p\epsilon$, in (54), then by using $\sigma {\varsigma }^{1/p}\zeta =\sigma {{\varsigma }^p\epsilon +{\varsigma }^p\upsilon -\varsigma \zeta }^{1/p}$ and $k$-fractional integral operators (6) and (7), the second inequality of (51) is acquired.

(ii) Proof is the same as the proof of (i)

Corollary 23.

By utilizing (51) and (52), some more $k$-fractional inequalities are offered as noted below:

(i) By choosing $\varsigma =I$ and $\begin{array}{cc}\tilde{p}=0,& p=-1\end{array}$, we acquire

$$\begin{array}{}\text{(55)}& \sigma \frac{2\epsilon \upsilon }{\epsilon +\upsilon }{\int }_{2\epsilon \upsilon /\epsilon +\upsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{\frac{1}{\upsilon }-\zeta }^{\mu }r{\zeta }^{-1}d\zeta \le h\frac{1}{2}{\int }_{2\epsilon \upsilon /\epsilon +\upsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{\frac{1}{\upsilon }-\zeta }^{\mu }\sigma {\zeta }^{-1}r{\zeta }^{-1}d\zeta +{\int }_{1/\epsilon }^{2\epsilon \upsilon /\epsilon +\upsilon }{\zeta -\frac{1}{\epsilon }}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{\zeta -\frac{1}{\epsilon }}^{\mu }\sigma {\zeta }^{-1}r{\zeta }^{-1}d\zeta \le h\frac{1}{2}\sigma \epsilon +\sigma \upsilon {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu }r\frac{2\epsilon \upsilon }{t\upsilon +2-t\epsilon }h\frac{t}{2}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu }r\frac{2\epsilon \upsilon }{t\upsilon +2-t\epsilon }h\frac{2-t}{2}dt\end{array}$$

(ii) By choosing $p=-1$ and $\varsigma =I$, we acquire

$$\begin{array}{}\text{(56)}& \sigma \frac{2\epsilon \upsilon }{\epsilon +\upsilon }{\int }_{2\epsilon \upsilon /\epsilon +\upsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{\frac{1}{\upsilon }-\zeta }^{\mu };\tilde{p}r{\zeta }^{-1}d\zeta \le h\frac{1}{2}{\int }_{2\epsilon \upsilon /\epsilon +\upsilon }^{1/\upsilon }{\frac{1}{\upsilon }-\zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{\frac{1}{\upsilon }-\zeta }^{\mu };\tilde{p}\sigma {\zeta }^{-1}r{\zeta }^{-1}d\zeta +{\int }_{1/\epsilon }^{2\epsilon \upsilon /\epsilon +\upsilon }{\zeta -\frac{1}{\epsilon }}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{\zeta -\frac{1}{\epsilon }}^{\mu };\tilde{p}\sigma {\zeta }^{-1}r{\zeta }^{-1}d\zeta \le h\frac{1}{2}\sigma \epsilon +\sigma \upsilon {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r\frac{2\epsilon \upsilon }{t\upsilon +2-t\epsilon }h\frac{t}{2}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r\frac{2\epsilon \upsilon }{t\upsilon +2-t\epsilon }h\frac{2-t}{2}dt\end{array}$$

(iii) By choosing $p=-1$, we acquire

$$\begin{array}{}\text{(57)}& \sigma {\frac{{\varsigma }^{-1}\epsilon +{\varsigma }^{-1}\upsilon }{2}}^{-1}\times {\int }_{{\varsigma }^{-1}{\varsigma }^{-1}\epsilon +{\varsigma }^{-1}\upsilon /2}^{{\varsigma }^{-1}{\varsigma }^{-1}\upsilon }{{\varsigma }^{-1}\upsilon -\varsigma \zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{{\varsigma }^{-1}\upsilon -\varsigma \zeta }^{\mu };\tilde{p}r{\varsigma }^{-1}\zeta {\varsigma }^{\prime }\zeta d\zeta \le h\frac{1}{2}{\int }_{{\varsigma }^{-1}{\varsigma }^{-1}\epsilon +{\varsigma }^{-1}\upsilon /2}^{{\varsigma }^{-1}{\varsigma }^{-1}\upsilon }{{\varsigma }^{-1}\upsilon -\varsigma \zeta }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{{\varsigma }^{-1}\upsilon -\varsigma \zeta }^{\mu };\tilde{p}\sigma {\varsigma }^{-1}\zeta r{\varsigma }^{-1}\zeta {\varsigma }^{\prime }\zeta d\zeta +{\int }_{{\varsigma }^{-1}{\varsigma }^{-1}\epsilon }^{{\varsigma }^{-1}{\varsigma }^{-1}\epsilon +{\varsigma }^{-1}\upsilon /2}{\varsigma \zeta -{\varsigma }^{-1}\epsilon }^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}\overline{\overline{w}}{\varsigma \zeta -{\varsigma }^{-1}\epsilon }^{\mu };\tilde{p}\sigma {\varsigma }^{-1}\zeta r{\varsigma }^{-1}\zeta {\varsigma }^{\prime }\zeta d\zeta \le h\frac{1}{2}\sigma \varsigma \epsilon +\sigma \varsigma \upsilon {\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r\frac{2\varsigma \epsilon \varsigma \upsilon }{t\varsigma \upsilon +2-t\varsigma \epsilon }h\frac{t}{2}dt+{\int }_0^1{t}^{\alpha /k-1}{E}_{\mu ,\alpha ,l}^{\gamma ,\delta ,v,c}w{t}^{\mu };\tilde{p}r\frac{2\varsigma \epsilon \varsigma \upsilon }{t\varsigma \upsilon +2-t\varsigma \epsilon }h\frac{2-t}{2}dt\end{array}$$

Remark 24.

The well-known $k$-fractional inequalities are further noted as below:

(i) By choosing $ht=t$ and $k=1$ in Corollary 23 (i), Theorem 2.6 of [24] is acquired

(ii) By choosing $ht=t$ and $k=1$ in Corollary 23 (ii), Theorem 2.6 of [5] is acquired

(iii) By choosing $ht=t$ and $k=1$ in Corollary 23 (iii), Theorem 2.10 of [27] is acquired

3. Conclusion

In this article, generalized versions of $k$-fractional Hadamard and Fejér-Hadamard inequalities are presented. To obtain these inequalities, $k$-fractional integral operators including the Mittag-Leffler function have been used for $p,h$-convex functions. Many published results in the literature are directly connected with the findings of this paper. Some corollaries have been formulated as new $k$-fractional Hadamard and Fejér-Hadamard inequalities.

Acknowledgments

The study was funded by the Science & Technology Bureau of Chengdu 2020-YF09-00005-SN supported by the Sichuan Science and Technology program 2021YFH0107 Erasmus+ SHYFTE Project 598649-EPP-1-2018-1-FR-EPPKA2-CBHE-JP.

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