1. Introduction
Convex functions are very important in the field of mathematical inequalities. Nobody can deny the importance of convex functions. A large number of mathematical inequalities exist in literature due to convex functions. For more information related to convex functions and it’s properties (see, [
1,
2,
3]).
Definition 1.
A function \(\mu:I\rightarrow \mathbb{R}\) on an interval of real line is said to be convex, if for all \(\alpha,\beta\in I\) and \(\kappa\in[0,1]\), the following inequality holds:
\begin{equation}\label{34}
\mu(\kappa \alpha+(1-\kappa)\beta)\leq \kappa\mu(\alpha)+(1-\kappa)\mu(\beta).
\end{equation}
(1)
The function \(\mu\) is said to be concave if \(-\mu\) is convex.
A convex function is interpreted very nicely in the coordinate plane by the well known Hadamard inequality stated as follows:
Theorem 2.
Let \(\mu:[\alpha,\beta]\rightarrow \mathbb{R}\) be a convex function such that \(\alpha< \beta\). The following inequalities holds:
\begin{equation*}
\mu\left(\frac{\alpha+\beta}{2}\right)\leq \frac{1}{\beta-\alpha}\int^{\beta}_{\alpha}\mu(\kappa)d\kappa\leq \frac{\mu(\alpha)+\mu(\beta)}{2}.
\end{equation*}
In [
4], Fejér gave the generalization of Hadamard inequality known as the Fejér-Hadamard inequality stated as follows:
Theorem 3.
Let \(\mu:[\alpha,\beta]\rightarrow \mathbb{R}\) be a convex function such that \(\alpha< \beta\). Also let \(\nu:[\alpha,\beta]\rightarrow \mathbb{R}\) be a positive, integrable and symmetric to \(\frac{\alpha+\beta}{2}\). The following inequalities hold:
\begin{equation}\label{r} \mu\left(\frac{\alpha+\beta}{2}\right)\int_{\alpha}^{\beta}\nu(\kappa)d\kappa\leq \int_{\alpha}^{\beta}\mu(\kappa)\nu(\kappa)d\kappa\leq \frac{\mu(\alpha)+\mu(\beta)}{2}\int_{\alpha}^{\beta}\nu(\kappa)d\kappa.
\end{equation}
(2)
The Hadamard and the Fejér-Hadamard inequalities are further generalized in various ways by using different fractional integral operators such as Riemann-Liouville, Katugampola, conformable and generalized fractional integral operators containing Mittag-Leffler function etc. For more results and details (see, [
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21]).
Next we give the definition of exponentially convex functions.
Definition 4.[9, 22]
A function \(\mu: I\rightarrow\mathbb{R}\) on an interval of real line is said to be exponentially convex, if for all \(\alpha,\beta \in I\) and \(\kappa\in[0,1]\), the following inequality holds:
\begin{equation}\label{11}
e^{ \mu(\kappa \alpha+(1-\kappa)\beta)}\leq \kappa e^{\mu(\alpha)}+(1-\kappa)e^{\mu(\beta)}.
\end{equation}
(3)
In [
23], Rashid
et al., gave the definition of exponentially \(s\)-convex functions.
Definition 5. Let \(s\in[0,1]\). A function \(\mu: I\rightarrow\mathbb{R}\) on an interval of real line is said to be exponentially \(s\)-convex, if for all \(\alpha,\beta \in I\) and \(\kappa\in[0,1]\), the following inequality holds:
\begin{equation}\label{12}
e^{ \mu(\kappa \alpha+(1-\kappa)\beta)}\leq \kappa^s e^{\mu(\alpha)}+(1-\kappa)^se^{\mu(\beta)}.
\end{equation}
(4)
In [
24], Rashid
et al., gave the definition of exponentially \(h\)-convex functions.
Definition 6. Let \(J\subseteq\mathbb{R}\) be an interval containing \((0,1)\) and let \(h: J\rightarrow\mathbb{R}\) be a non-negative function. Then a function \(\mu: I\rightarrow\mathbb{R}\) on an interval of real line is said to be exponentially \(h\)-convex, if for all \(\alpha,\beta \in I\) and \(\kappa\in[0,1]\), the following inequality holds:
\begin{equation}\label{13}
e^{ \mu(\kappa \alpha+(1-\kappa)\beta)}\leq h(\kappa) e^{\mu(\alpha)}+h(1-\kappa)e^{\mu(\beta)}.
\end{equation}
(5)
In [
25], Rashid
et al., gave the definition of exponentially \(m\)-convex functions.
Definition 7. A function \(\mu: I\rightarrow\mathbb{R}\) on an interval of real line is said to be exponentially \(m\)-convex, if for all \(\alpha,\beta \in I\), \(m\in(0,1]\) and \(\kappa\in[0,1]\), the following inequality holds:
\begin{equation}\label{14}
e^{ \mu(\kappa \alpha+m(1-\kappa)\beta)}\leq \kappa e^{\mu(\alpha)}+m(1-\kappa)e^{\mu(\beta)}.
\end{equation}
(6)
In [
26], Rashid
et al., gave the definition of exponentially \((h,m)\)-convex functions.
Definition 8. Let \(J\subseteq\mathbb{R}\) be an interval containing \((0,1)\) and let \(h: J\rightarrow\mathbb{R}\) be a non-negative function. Then a function \(\mu: I\rightarrow\mathbb{R}\) on an interval of real line is said to be exponentially \((h,m)\)-convex, if for all \(\alpha,\beta \in I\), \(m\in(0,1]\) and \(\kappa\in[0,1]\), the following inequality holds:
\begin{equation}\label{15}
e^{ \mu(\kappa \alpha+m(1-\kappa)\beta)}\leq h(\kappa) e^{\mu(\alpha)}+mh(1-\kappa)e^{\mu(\beta)}.
\end{equation}
(7)
Remark 1.
- If we set \(h(\kappa)=\kappa\) and \(m=1\) in (7), then exponentially convex function (3) is obtained.
- If we set \(h(\kappa)=\kappa^s\) and \(m=1\) in (7), then exponentially \(s\)-convex function (4) is obtained.
- If we set \(m=1\) in (7), then exponentially \(h\)-convex function (5) is obtained.
- If we set \(h(\kappa)=\kappa\) in (7), then exponentially \(m\)-convex function (6) is obtained.
Fractional integral operators also play important role in the subject of mathematical analysis. Recently in [
27], Andrić
et al., defined the generalized fractional integral operators containing generalized Mittag-Leffler function in their kernels as follows:
Definition 9. Let \(\psi,\sigma,\phi,l,\varsigma,c\in \mathbb{C}\), \(\Re(\sigma),\Re(\phi),\Re(l)>0\), \(\Re(c)>\Re(\varsigma)>0\) with \(p\geq0\), \(r>0\) and \(0< q\leq r+\Re(\sigma)\). Let \(\mu\in L_{1}[\alpha,\beta]\) and \(u\in[\alpha,\beta].\) Then the generalized fractional integral operators \(\Upsilon_{\sigma,\phi,l,\psi,\alpha^{+}}^{\varsigma,r,q,c}\mu \) and \(\Upsilon_{\sigma,\phi,l,\psi,\beta^{-}}^{\varsigma,r,q,c}\mu\) are defined by:
\begin{equation}\label{a}
\left(\Upsilon_{\sigma,\phi,l,\psi,\alpha^{+}}^{\varsigma,r,q,c}\mu \right)(u;p)=\int_{\alpha}^{u}(u-\kappa)^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi(u-\kappa)^{\sigma};p)\mu(\kappa)d\kappa,
\end{equation}
(8)
\begin{equation}
\left(\Upsilon_{\sigma,\phi,l,\psi,\beta^{-}}^{\varsigma,r,q,c}\mu \right)(u;p)=\int_{u}^{\beta}(\kappa-u)^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi(\kappa-u)^{\sigma};p)\mu(\kappa)d\kappa,
\end{equation}
(9)
where \(E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\kappa;p)\) is the generalized Mittag-Leffler function defined as follows:
\begin{equation*}
E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\kappa;p)= \sum_{n=0}^{\infty}\frac{\beta_{p}(\varsigma+nq,c-\varsigma)}{\beta(\varsigma,c-\varsigma)} \frac{(c)_{nq}}{\Gamma(\sigma n +\phi)} \frac{\kappa^{n}}{(l)_{n r}}.
\end{equation*}
In [
28], Farid defined the following unified integral operators:
Definition 10.
Let \(\mu, \nu: [\alpha,\beta]\rightarrow \mathbb{R}\), \(0< \alpha0\), \(\Re(c)>\Re(\varsigma)>0\) with \(p\geq0\), \(\sigma,r>0\) and \(0< q\leq r+\sigma\). Then for \(u\in[\alpha,\beta]\) the integral operators \(_{\nu}\Upsilon_{\sigma, \phi,l, \alpha^{+}}^{\gamma, \varsigma,r,q,c}\mu\) and \(_{\nu}\Upsilon_{\sigma, \phi,l, \beta^{-}}^{\gamma, \varsigma,r,q,c}\mu\) are defined by:
\begin{align}\label{sd}
\left(_{\nu}\Upsilon_{\sigma, \phi,l, \alpha^{+}}^{\gamma, \varsigma,r,q,c}\mu\right)(u;p)=\int_{\alpha}^{u}\frac{\gamma(\nu(u)-\nu(\kappa))}{\nu(u)-\nu(\kappa)} E_{\sigma, \phi, l}^{\varsigma,r,q,c} (\psi(\nu(u)-\nu(\kappa))^{\sigma};p)\mu(\kappa)d(\nu(\kappa)),
\end{align}
(10)
\begin{align}\label{sb}
\left(_{\nu}\Upsilon_{\sigma, \phi,l, \beta^{-}}^{\gamma, \varsigma,r,q,c}\mu\right)(u;p)=\int_{u}^{\beta}\frac{\gamma(\nu(\kappa)-\nu(u))}{\nu(\kappa)-\nu(u)} E_{\sigma, \phi, l}^{\varsigma,r,q,c} (\psi(\nu(\kappa)-\nu(u))^{\sigma};p)\mu(\kappa)d(\nu(\kappa)).
\end{align}
(11)
If we set \(\gamma(u)=u^\phi\) in (10) and (11), then we get the following generalized fractional integral operators containing Mittag-Leffler function:
Definition 11.
Let \(\mu, \nu: [\alpha,\beta]\rightarrow \mathbb{R}\), \(0< \alpha0\), \(\Re(c)>\Re(\varsigma)>0\) with \(p\geq0\), \(\sigma,r>0\) and \(0< q\leq r+\sigma\). Then for \(u\in[\alpha,\beta]\) the integral operators \(_{\nu}\Upsilon_{\sigma, \phi,l,\psi, \alpha^{+}}^{\varsigma,r,q,c}\mu\) and \(_{\nu}\Upsilon_{\sigma, \phi,l,\psi, \beta^{-}}^{ \varsigma,r,q,c}\mu\) are defined by:
\begin{equation}\label{1}
\left(_{\nu}\Upsilon_{\sigma, \phi,l, \psi, \alpha^{+}}^{ \varsigma,r,q,c}\mu\right)(u;p)=\int_{\alpha}^{u}(\nu(u)-\nu(\kappa))^{\phi-1} E_{\sigma, \phi, l}^{\varsigma,r,q,c} (\psi(\nu(u)-\nu(\kappa))^{\sigma};p)\mu(\kappa)d(\nu(\kappa)),
\end{equation}
(12)
\begin{equation}\label{7}
\left(_{\nu}\Upsilon_{\sigma, \phi,l,\psi, \beta^{-}}^{ \varsigma,r,q,c}\mu\right)(u;p)=\int_{u}^{\beta}(\nu(\kappa)-\nu(u))^{\phi-1} E_{\sigma, \phi, l}^{\varsigma,r,q,c} (\psi(\nu(\kappa)-\nu(u))^{\sigma};p)\mu(\kappa)d(\nu(\kappa)).
\end{equation}
(13)
Remark 2.
(12) and (13) are the generalization of the following fractional integral operators:
- Setting \(\nu(u)=u\), the fractional integral operators (8) and (9), can be obtained.
- Setting \(\nu(u)=u\) and \(p=0\), the fractional integral operators defined by Salim-Faraj in [29]TOS}, can be obtained.
- Setting \(\nu(u)=u\) and \(l=r=1\), the fractional integral operators defined by Rahman et al., in [30], can be obtained.
- Setting \(\nu(u)=u\), \(p=0\) and \(l=r=1\), the fractional integral operators defined by Srivastava-Tomovski in [31], can be obtained.
- Setting \(\nu(u)=u\), \(p=0\) and \(l=r=q=1\), the fractional integral operators defined by Prabhakar in [32], can be obtained.
- Setting \(\nu(u)=u\) and \(\psi=p=0\), the Riemann-Liouville fractional integral operators can be obtained.
In [
33], Mehmood
et al., proved the following formulas for constant function:
\begin{equation}\label{1*}
\left(_{\nu}\Upsilon_{\sigma, \phi,l, \psi, \alpha^{+}}^{ \varsigma,r,q,c}1\right)(u;p)=(\nu(u)-\nu(\alpha))^{\phi}E_{\sigma, \phi+1, l}^{\varsigma,r,q,c} (\psi(\nu(u)-\nu(\alpha))^{\sigma};p):=_{\nu}\xi_{\psi, \alpha^{+}}^{\phi}(u;p),
\end{equation}
(14)
\begin{equation}\label{7*}
\left(_{\nu}\Upsilon_{\sigma, \phi,l, \psi, \beta^{-}}^{ \varsigma,r,q,c}1\right)(u;p)=(\nu(\beta)-\nu(u))^{\phi}E_{\sigma, \phi+1, l}^{\varsigma,r,q,c} (\psi(\nu(\beta)-\nu(u))^{\sigma};p):=_{\nu}\xi_{\psi, \beta^{-}}^{ \phi}(u;p).
\end{equation}
(15)
The objective of this paper is to establish the Hadamard and the Fejér-Hadamard inequalities for generalized fractional integral operators (12) and (13) containing Mittag-Leffler function via a monotone function by using the exponentially \((h,m)\)-convex functions. These inequalities lead to produce the Hadamard and the Fejér-Hadamard inequalities for various kinds of exponentially convexity and well known fractional integral operators given in Remark 1 and Remark 2. In Section 2, we prove the Hadamard inequalities for generalized fractional integral operators (12) and (13) via exponentially \((h,m)\)-convex functions. In Section 3, we prove the Fejér-Hadamard inequalities for these generalized fractional integral operators via exponentially \((h,m)\)-convex functions. Moreover, some of the results published in [
26,
33,
34] have been obtained in particular.
2.Fractional Hadamard inequalities for exponentially \((h,m)\)-convex functions
In this section, we will give two versions of the generalized fractional Hadamard inequality. To establish these inequalities exponentially \((h,m)\)-convexity and generalized fractional integrals operators have been used.
Theorem 12.
Let \(\mu,\nu: [\alpha,m\beta]\subset[0,\infty)\to\mathbb{R}\), \(0< \alpha< m\beta\) be two functions such that \(\mu\) be integrable and \(\nu\) be differentiable. If \(\mu\) be exponentially \((h,m)\)-convex, \(\nu\) be strictly increasing and \(h\in[0,1]\). Then for generalized fractional integral operators, the following inequalities hold:
\begin{align}\label{yty}
&{e^{\mu\left( \frac{\nu(\alpha)+m\nu(\beta)}{2}\right)}} _{\nu}\xi^{\phi}_{\bar{\psi},\alpha^{+}}(\nu^{-1}(m\nu(\beta));p)\nonumber\\\nonumber
&\leq\! h\!\left(\!\frac{1}{2}\!\right)\!\!\left[\!\left(\!_{\nu}\Upsilon_{\sigma,\phi,l,\bar{\psi},\alpha^{+}}^{\varsigma,r,q,c}e^{\mu\circ\nu}\!\right)\!\!(\!\nu^{-1}(\!m\nu(\beta));p)\!+\!\nonumber m^{\phi+1}\!\!\left(\!_{\nu}\Upsilon_{\sigma,\phi,l,\bar{\psi}m^\sigma,\beta^{-}}^{\varsigma,r,q,c}e^{\mu\circ\nu}\!\!\right)\!\!\left(\!\!\nu^{-1}\!\left(\!\frac{\nu(\alpha)}{m}\!\right)\!;p\!\right)\!\right]\\\nonumber
&\leq h\left(\frac{1}{2}\right){(m\nu(\beta)-\nu(\alpha))}^\phi\left[\left(e^{\mu(\nu(\alpha))}+me^{\mu(\nu(\beta))}\right) \left(\Upsilon_{\sigma,\phi,l,{\psi},1^{-}}^{\varsigma,r,q,c}h\right)(0;p)\right.\nonumber\\
&\;\;\;\left.+m\left(e^{\mu(\nu(\beta))}+me^{\mu\left(\frac{\nu(\alpha)}{m^2}\right)}\right)\left(\Upsilon_{\sigma,\phi,l,{\psi},0^{+}}^{\varsigma,r,q,c}h\right)(1;p)\right],\;\;\text{where}\;\; \bar{\psi}=\frac{\psi}{(m\nu(\beta)-\nu(\alpha))^{\sigma}}.
\end{align}
(16)
Proof.
By the exponentially \((h,m)\)-convexity of \(\mu\), we have
\begin{equation}\label{b}
{e^{\mu\left( \frac{\nu(\alpha)+m\nu(\beta)}{2}\right)}}\leq h\left(\frac{1}{2}\right) \left[{e^{{\mu(\kappa\nu(\alpha)+m(1-\kappa)\nu(\beta))}}}+m{e^{{\mu((1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta))}}}\right].
\end{equation}
(17)
Multiplying (17) with \(\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)\) and integrating over \([0,1]\), we have
\begin{align}\label{ad}
&{e^{\mu\left( \frac{\nu(\alpha)+m\nu(\beta)}{2}\right)}}\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)d\kappa\notag
\\&\leq h\left(\frac{1}{2}\right)
\left[ \int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p){e^{{\mu(\kappa\nu(\alpha)+m(1-\kappa)\nu(\beta))}}}d\kappa\right.\left.+m\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p){e^{{\mu((1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta))}}}d\kappa\right].
\end{align}
(18)
Setting \(\nu(u)=\kappa\nu(\alpha)+m(1-\kappa)\nu(\beta)\) and \(\nu(v)=(1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta)\) in (18), then again from exponentially \((h,m)\)-convexity of \(\mu\), we have
\begin{align}\label{c}
&e^{{\mu(\kappa\nu(\alpha)+m(1-\kappa)\nu(\beta))}}+me^{{\mu((1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta))}}\notag\\
&\leq h(\kappa)\left(e^{\mu(\nu(\alpha))}+me^{\mu(\nu(\beta))}\right)+mh(1-\kappa)\left(e^{\mu(\nu(\beta))}+me^{\mu\left(\frac{\nu(\alpha)}{m^2}\right)}\right).
\end{align}
(19)
Multiplying (19) with \(h\left(\frac{1}{2}\right)\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)\) and integrating over \([0,1]\), we have
\begin{align}\label{ab}
&\notag h\left(\frac{1}{2}\right)\bigg[\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)e^{{\mu(\kappa\nu(\alpha)+m(1-\kappa)\nu(\beta))}}d\kappa+m\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)e^{{\mu((1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta))}}d\kappa\bigg]\\&\leq h\left(\frac{1}{2}\right) \bigg[\left(e^{\mu(\nu(\alpha))}+me^{\mu(\nu(\beta))}\right)\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)h(\kappa)d\kappa+m\left(e^{\mu(\nu(\beta))}+me^{\mu\left(\frac{\nu(\alpha)}{m^2}\right)}\right) \nonumber\\&\;\;\;\times \int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)h(1-\kappa)d\kappa\bigg].
\end{align}
(20)
Setting \(\nu(u)=\kappa\nu(\alpha)+m(1-\kappa)\nu(\beta)\) and \(\nu(v)=(1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta)\) in (20), then by using (8), (9), (12)and (13), the second inequality of (16) is obtained.
Corollary 1.
Setting \(m=1\) in (16), the following inequalities for exponentially \(h\)-convex function can be obtained:
\begin{align}\label{yty*}
& {e^{\mu\left( \frac{\nu(\alpha)+\nu(\beta)}{2}\right)}} _{\nu}\xi^{\phi}_{\bar{\psi},\alpha^{+}}(\beta;p)\leq h\left(\frac{1}{2}\right)\bigg[\left(_{\nu}\Upsilon_{\sigma,\phi,l,\bar{\psi},\alpha^{+}}^{\varsigma,r,q,c}e^{\mu\circ\nu}\right)(\beta;p)+ \left(_{\nu}\Upsilon_{\sigma,\phi,l,\bar{\psi},\beta^{-}}^{\varsigma,r,q,c}e^{\mu\circ\nu}\right)\left(\alpha;p \right)\bigg]\nonumber\\
&\leq h\left(\frac{1}{2}\right){(\nu(\beta)-\nu(\alpha))}^\phi\left(e^{\mu(\nu(\alpha))}+e^{\mu(\nu(\beta))}\right)\left[ \left(\Upsilon_{\sigma,\phi,l,{\psi},1^{-}}^{\varsigma,r,q,c}h\right)(0;p)+\left(\Upsilon_{\sigma,\phi,l,{\psi},0^{+}}^{\varsigma,r,q,c}h\right)(1;p)\right],
\end{align}
(21)
where \(\bar{\psi}=\frac{\psi}{(\nu(\beta)-\nu(\alpha))^{\sigma}}. \)
Remark 3.
- If we set \(h(\kappa)=\kappa\) in (16), then [33, Theorem 8] is obtained.
- If we set \(h(\kappa)=\kappa\) and \(m=1\) in (16), then [33, Corollary 1] is obtained.
- If we set \(\nu(u)=u\) and \(h(\kappa)=\kappa\) in (16), then [34, Theorem 2.1] is obtained.
- If we set \(\nu(u)=u\), \(h(\kappa)=\kappa\) and \(m=1\) in (16), then [34, Corollary 2.2] is obtained.
- If we set \(\nu(u)=u\) in (16), then [26, Theorem 2.1] is obtained.
In the following we give another version of the Hadamard inequality for generalized fractional integral operators via exponentially \((h,m)\)-convex functions.
Theorem 13.
Let \(\mu,\nu: [\alpha,m\beta]\subset[0,\infty)\to\mathbb{R}\), \(0< \alpha< m\beta\) be two functions such that \(\mu\) be integrable and \(\nu\) be differentiable. If \(\mu\) be exponentially \((h,m)\)-convex and \(\nu\) be strictly increasing. Then for generalized fractional integral operators, the following inequalities hold:
\begin{align}\label{lmo}
&\notag{e^{\mu\left( \frac{\nu(\alpha)+m\nu(\beta)}{2}\right)}}_{\nu}\xi^{\phi}_{\bar{\psi}2^\sigma,\left(\nu^{-1}\left(\frac{\nu(\alpha)+m\nu(\beta)}{2}\right)\right) ^{+}}(\nu^{-1}(m\nu(\beta));p)\\\nonumber
&\leq h\!\left(\frac{1}{2}\right)\! \left[\left(_{\nu}\Upsilon^{\varsigma,r,q,c}_{\sigma,\phi,l,\bar{\psi}2^\sigma,\left(\nu^{-1}\left(\frac{\nu(\alpha)+m\nu(\beta)}{2}\right)\right) ^{+}}e^{\mu\circ\nu}\right)(\nu^{-1}(m\nu(\beta));p)\right.\\&\nonumber\;\;\;\left.+m^{\phi+1} \left(_{\nu}\Upsilon^{\varsigma,r,q,c}_{\sigma,\phi,l,\bar{\psi}(2m)^\sigma,\left(\nu^{-1}\left(\frac{\nu(\alpha)+m\nu(\beta)}{2m}\right)\right) ^{-}}e^{\mu\circ\nu}\right)\left(\nu^{-1}\left(\frac{\nu(\alpha)}{m}\right);p \right)\right]
\\&\nonumber\leq\!h\!\left(\!\frac{1}{2}\!\right)\!\! \frac{(m\nu(\beta)-\nu(\alpha))^\phi}{2^{\phi}}\!\!\left[\!\left(\!e^{\mu(\nu(\alpha))}+me^{\mu(\nu(\beta))}\!\right)\!\!\int_{0}^{1}\!\!\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)h\left(\frac{\kappa}{2}\right)\!d\kappa\right.\\&\;\;\;\left.+m\!\left(\!e^{\mu(\nu(\beta))}+me^{\mu\left(\frac{\nu(\alpha)}{m^2}\!\right)}\right)\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)h\left(\frac{2-\kappa}{2}\right)d\kappa\right],
\end{align}
(22)
where \(\bar{\psi}\) is same as in (16).
Proof.
By the exponentially \((h,m)\)-convexity of \(\mu\), we have
\begin{align}\label{d}
{e^{\mu\left( \frac{\nu(\alpha)+m\nu(\beta)}{2}\right)}} \leq h\left(\frac{1}{2}\right)\left[{e^{\mu\left( \frac{\kappa}{2}\nu(\alpha)+m\frac{(2-\kappa)}{2}\nu(\beta)\right)}}+m
{e^{\mu\left(\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\right)}}\right].
\end{align}
(23)
Multiplying (23) with \(\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)\) and integrating over \([0,1]\), we have
\begin{align}\label{w}
&{e^{\mu\left( \frac{\nu(\alpha)+m\nu(\beta)}{2}\right)}}\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)d\kappa \nonumber\\
&\leq h\left(\frac{1}{2}\right) \bigg[\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p){e^{\mu\left( \frac{\kappa}{2}\nu(\alpha)+m\frac{(2-\kappa)}{2}\nu(\beta)\right)}}d\kappa+m\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p) {e^{\mu\left(\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\right)}}d\kappa\bigg].
\end{align}
(24)
Setting \(\nu(u)=\frac{\kappa}{2}\nu(\alpha)+m\frac{(2-\kappa)}{2}\nu(\beta)\) and \(\nu(v)=\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\) in (24), then
by using (12), (13) and (14), the first inequality of (22) is obtained.
Again from exponentially \((h,m)\)-convexity of \(\mu\), we have
\begin{align}\label{e}
&e^{\mu\left( \frac{\kappa}{2}\nu(\alpha)+m\frac{(2-\kappa)}{2}\nu(\beta)\right)} +me^{\mu\left(\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\right)} \\\nonumber
&\leq h\left(\frac{\kappa}{2}\right)\left(e^{\mu(\nu(\alpha))}+me^{\mu(\nu(\beta))}\right)+mh\left(\frac{2-\kappa}{2}\right)\left(e^{\mu(\nu(\beta))}+me^{\mu\left(\frac{\nu(\alpha)}{m^2}\right)}\right)\nonumber.
\end{align}
(25)
Multiplying (25) with \(h\left(\frac{1}{2}\right) \kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)\) and integrating over \([0,1]\), we have
\begin{align}\label{x}
&h\left(\frac{1}{2}\right)\bigg[\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)e^{\mu\left( \frac{\kappa}{2}\nu(\alpha)+m\frac{(2-\kappa)}{2}\nu(\beta)\right)}d\kappa+m\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)e^{\mu\left(\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\right)}d\kappa\nonumber\bigg]\\&\nonumber\leq h\left(\frac{1}{2}\right)
\bigg[ \left(e^{\mu(\nu(\alpha))}+me^{\mu(\nu(\beta))}\right)\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)h\left(\frac{\kappa}{2}\right)d\kappa\nonumber\\&\;\;\;+m\left(e^{\mu(\nu(\beta))}+me^{\mu\left(\frac{\nu(\alpha)}{m^2}\right)}\right) \int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)h\left(\frac{2-\kappa}{2}\right)d\kappa\bigg].
\end{align}
(26)
Putting \(\nu(u)=\frac{\kappa}{2}\nu(\alpha)+m\frac{(2-\kappa)}{2}\nu(\beta)\) and \(\nu(v)=\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\) in (26), then
by using (12) and (13), the second inequality of (22) is obtained.
Corollary 2.
Setting \(m=1\) in (22), the following inequalities for exponentially \(h\)-convex function can be obtained:
\begin{align}\label{lmo*}
&\nonumber2{e^{\mu\left( \frac{\nu(\alpha)+\nu(\beta)}{2}\right)}}_{\nu}\xi^{\phi}_{\bar{\psi}2^\sigma,\left(\nu^{-1}\left(\frac{\nu(\alpha)+\nu(\beta)}{2}\right)\right) ^{+}}(\beta;p)\\\nonumber
&\leq h\left(\frac{1}{2}\right) \left[\left(_{\nu}\Upsilon^{\varsigma,r,q,c}_{\sigma,\phi,l,\bar{\psi}2^\sigma,\left(\nu^{-1}\left(\frac{\nu(\alpha)+\nu(\beta)}{2}\right)\right) ^{+}}e^{\mu\circ\nu}\right)(\beta;p)\right.\left.+ \left(_{\nu}\Upsilon^{\varsigma,r,q,c}_{\sigma,\phi,l,\bar{\psi}2^\sigma,\left(\nu^{-1}\left(\frac{\nu(\alpha)+\nu(\beta)}{2}\right)\right) ^{-}}e^{\mu\circ\nu}\right)(\alpha;p)\right]\\
&\leq h\left(\frac{1}{2}\right) \frac{(\nu(\beta)-\nu(\alpha))^\phi}{2^{\phi}}\left(\!e^{\mu(\nu(\alpha))}+e^{\mu(\nu(\beta))}\right)\!\left[\!\int_{0}^{1}\!\!\!\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)h\left(\frac{\kappa}{2}\right)\!d\kappa\!+\!\!\!\int_{0}^{1}\!\!\!\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)h\left(\!\frac{2-\kappa}{2}\!\right)\!d\kappa\right]
\end{align}
(27)
where \(\bar{\psi}\) is same as in (21).
Remark 4.
- If we set \(h(\kappa)=\kappa\) in (22), then [33, Theorem 9] is obtained.
- If we set \(h(\kappa)=\kappa\) and \(m=1\) in (22), then [33, Corollary 2] is obtained.
- If we set \(\nu(u)=u\) and \(h(\kappa)=\kappa\) in (22), then [34, Theorem 2.4] is obtained.
- If we set \(\nu(u)=u\), \(h(\kappa)=\kappa\) and \(m=1\) in (22), then [34, Corollary 2.5] is obtained.
- If we set \(\nu(u)=u\) in (22), then [26, Theorem 2.2] is obtained.
3.Fractional Fejér-Hadamard Inequalities for exponentially \((h,m)\)-convex functions
In this section, we will give two versions of the generalized fractional Fejér-Hadamard inequality. To establish these inequalities exponentially \((h,m)\)-convexity and generalized fractional integrals operators have been used.
Theorem 14.
Let \(\mu,\nu: [\alpha,m\beta]\subset[0,\infty)\to\mathbb{R}\), \(0< \alpha< m\beta\) be two functions such that \(\mu\) be integrable and \(\nu\) be differentiable. If \(\mu\) be exponentially \((h,m)\)-convex and \(\mu(\nu(v))=\mu(\nu(\alpha)+m\nu(\beta)-m\nu(v))\) and \(\nu\) be strictly increasing. Also, let \(\gamma : [\alpha,m\beta]\to\mathbb{R}\) be a function which is non-negative and integrable. Then for generalized fractional integral operators, the following inequalities hold:
\begin{align}\label{yy}
&\notag {e^{\mu\left( \frac{\nu(\alpha)+m\nu(\beta)}{2}\right)}}
\left(_{\nu}\Upsilon^{\varsigma,r,q,c}_{\sigma,\phi,l,\bar{\psi}m^\sigma,\beta^{-}}e^{\gamma\circ\nu}\right)\left(\nu^{-1}\left(\frac{\nu(\alpha)}{m}\right);p\right) \leq {h\left(\frac{1}{2}\right)(1+m)\left(_{\nu}\Upsilon^{\varsigma,r,q,c}_{\sigma,\phi,l,\bar{\psi}m^\sigma,\beta^{-}}e^{\mu\circ\nu}e^{\gamma\circ\nu}\right)\left(\nu^{-1}\left(\frac{\nu(\alpha)}{m}\right);p\right)}\\\nonumber
&\leq h\left(\frac{1}{2}\right) \frac{(m\nu(\beta)-\nu(\alpha))^\phi}{m^\phi}\bigg[\left(e^{\mu(\nu(\alpha))}+me^{\mu(\nu(\beta))}\right)\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)e^{\gamma( (1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta))}h\left(\kappa\right)d\kappa\\
&\;\;\;+\!m\!\left(\!e^{\mu(\nu(\beta))}\!+\!me^{\mu\left(\frac{\nu(\alpha)}{m^2}\right)}\!\right) \!\!\int_{0}^{1}\!\!\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)e^{\gamma( (1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta))}h\left(1-\kappa\right)\!d\kappa\bigg],\,
\end{align}
(28)
where \(\bar{\psi}\) is same as in (16).
Proof.
Multiplying (17) with \(\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)e^{\gamma( (1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta))}\) and integrating over \([0,1]\), we have
\begin{align}\label{z}
&\notag{e^{\mu\left( \frac{\nu(\alpha)+m\nu(\beta)}{2}\right)}}\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)e^{\gamma( (1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta))} d\kappa\\\nonumber
& \leq h\left(\frac{1}{2}\right) \bigg[\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p){e^{{\mu(\kappa\nu(\alpha)+m(1-\kappa)\nu(\beta))}}}e^{\gamma( (1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta))} d\kappa\\
&\;\;\;+m\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p){e^{{\mu((1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta))}}}e^{\gamma( (1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta))}d\kappa\bigg].
\end{align}
(29)
Setting \(\nu(v)=(1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta)\) in (29), then by using (13) and assumption \(\mu(\nu(v))=\mu(\nu(\alpha)+m\nu(\beta)-m\nu(v))\), the first inequality of (28) is obtained.
Now multiplying (19) with \(h\left(\frac{1}{2}\right)\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)e^{\gamma( (1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta))}\) and integrating over \([0,1]\), we have
\begin{align}\label{23}
&\notag h\left(\frac{1}{2}\right)\bigg[\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)e^{{\mu(\kappa\nu(\alpha)+m(1-\kappa)\nu(\beta))}}e^{\gamma( (1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta))} d\kappa\\
\nonumber&\;\;\;+m\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)e^{{\mu((1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta))}}e^{\gamma( (1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta))}d\kappa\bigg]\\&\nonumber\leq\! h\!\left(\!\frac{1}{2}\!\right) \!\!\bigg[\!\!\left(\!e^{\mu(\nu(\alpha))}+me^{\mu(\nu(\beta))}\!\right)\!\!\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)e^{\gamma( (1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta))}h\left(\kappa\right)d\kappa\\
&\;\;\;+\!m\!\left(\!e^{\mu(\nu(\beta))}\!+\!me^{\mu\left(\frac{\nu(\alpha)}{m^2}\right)}\!\right) \!\!\int_{0}^{1}\!\!\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)e^{\gamma( (1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta))}h\left(1-\kappa\right)\!d\kappa\bigg].
\end{align}
(30)
Setting \(\nu(v)=(1-\kappa)\frac{\nu(\alpha)}{m}+\kappa\nu(\beta)\) in (30), then by using (13) and assumption \(\mu(\nu(v))=\mu(\nu(\alpha)+m\nu(\beta)-m\nu(v))\), the second inequality of (28) is obtained.
Corollary 3.
Setting \(m=1\) in (28), the following inequalities for exponentially \(h\)-convex function can be obtained:
\begin{align}\label{yy**}
&\notag{e^{\mu\left( \frac{\nu(\alpha)+\nu(\beta)}{2}\right)}}
\left(_{\nu}\Upsilon^{\varsigma,r,q,c}_{\sigma,\phi,l,\bar{\psi},\beta^{-}}e^{\gamma\circ\nu}\right)\left(\alpha;p\right) \leq {2 h\left(\frac{1}{2}\right)\left(_{\nu}\Upsilon^{\varsigma,r,q,c}_{\sigma,\phi,l,\bar{\psi},\beta^{-}}e^{\mu\circ\nu}e^{\gamma\circ\nu}\right)\left(\alpha;p\right)}\\\nonumber
&\leq h\left(\frac{1}{2}\right) {(\nu(\beta)-\nu(\alpha))^\phi}\left(e^{\mu(\nu(\alpha))}+e^{\mu(\nu(\beta))}\right)\bigg[\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)e^{\gamma( (1-\kappa){\nu(\alpha)}+\kappa\nu(\beta))}h\left(\kappa\right)d\kappa\\
&\;\;\;+ \int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)e^{\gamma( (1-\kappa){\nu(\alpha)}+\kappa\nu(\beta))}h\left(1-\kappa\right)d\kappa\bigg],
\end{align}
(31)
where \(\bar{\psi}\) is same as in (21).
Remark 5.
- If we set \(h(\kappa)=\kappa\) in (28), then [33, Theorem 10] is obtained.
- If we set \(h(\kappa)=\kappa\) and \(m=1\) in (28), then [33, Corollary 3] is obtained.
- If we set \(\nu(u)=u\) and \(h(\kappa)=\kappa\) in (28), then [34, Theorem 2.7] is obtained.
- If we set \(\nu(u)=u\), \(h(\kappa)=\kappa\) and \(m=1\) in (28), then [34, Corollary 2.8] is obtained.
- If we set \(\nu(u)=u\) in (28), then [26, Theorem 2.3] is obtained.
In the following we give another generalized fractional version of the Fejér-Hadamard inequality.
Theorem 15.
Let \(\mu,\nu: [\alpha,m\beta]\subset[0,\infty)\to\mathbb{R}\), \(0< \alpha< m\beta\) be two functions such that \(\mu\) be integrable and \(\nu\) be differentiable. If \(\mu\) be exponentially \((h,m)\)-convex and \(\mu(\nu(v))=\mu(\nu(\alpha)+m\nu(\beta)-m\nu(v))\) and \(\nu\) be strictly increasing. Also, let \(\gamma : [\alpha,m\beta]\to\mathbb{R}\) be a function which is non-negative and integrable. Then for generalized fractional integral operators, the following inequalities hold:
\begin{align}\label{yy*}
&\notag{e^{\mu\left( \frac{\nu(\alpha)+m\nu(\beta)}{2}\right)}}
\!\!\left(\!_{\nu}\Upsilon^{\varsigma,r,q,c}_{\sigma,\phi,l,\bar{\psi}(2m)^\sigma,\left(\nu^{-1}\left(\frac{\nu(\alpha)+m\nu(\beta)}{2m}\right)\right) ^{-}}e^{\gamma\circ\nu}\!\right)\!\left(\!\nu^{-1}\!\left(\frac{\nu(\alpha)}{m}\!\right);p\!\right) \\&\nonumber\leq \!h\!\left(\!\frac{1}{2}\!\right) {\!(1\!+\!m)\!\!\left(\!\!_{\nu}\Upsilon^{\varsigma,r,q,c}_{\sigma,\phi,l,\bar{\psi}(2m)^\sigma,\left(\nu^{-1}\left(\frac{\nu(\alpha)+m\nu(\beta)}{2m}\right)\right) ^{-}}e^{\mu\circ\nu}e^{\gamma\circ\nu}\!\!\right)\!\!\left(\!\!\nu^{-1}\!\left(\!\frac{\nu(\alpha)}{m}\right);p\!\right)}\\\nonumber
&\leq h\left(\frac{1}{2}\right) \frac{(m\nu(\beta)-\nu(\alpha))^\phi}{(2m)^\phi}\bigg[\left(e^{\mu(\nu(\alpha))}+me^{\mu(\nu(\beta))}\right)\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p) {e^{\gamma\left(\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\right)}}h\left(\frac{\kappa}{2}\right)d\kappa\\
&\;\;\;+\!m\!\!\left(\!e^{\mu(\nu(\beta))}\!+\!me^{\mu\left(\frac{\nu(\alpha)}{m^2}\!\right)}\right)\!\!\!\int_{0}^{1}\!\!\!\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p) {e^{\gamma\left(\!\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\!\right)}}h\!\left(\!\frac{2\!-\!\kappa}{2}\!\right)\!d\kappa\bigg],
\end{align}
(32)
where \(\bar{\psi}\) is same as in (16).
Proof.
Multiplying (23) with \(\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p) {e^{\gamma\left(\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\right)}}\) and integrating over \([0,1]\), we have
\begin{align}\label{z*}
&\notag{e^{\mu\left( \frac{\nu(\alpha)+m\nu(\beta)}{2}\right)}}\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p) {e^{\gamma\left(\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\right)}} d\kappa\\\nonumber
& \leq h\left(\frac{1}{2}\right) \bigg[\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p){e^{\mu\left( \frac{\kappa}{2}\nu(\alpha)+m\frac{(2-\kappa)}{2}\nu(\beta)\right)}} {e^{\gamma\left(\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\right)}} d\kappa\\
&\;\;\;+m\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p) {e^{\mu\left(\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\right)}} {e^{\gamma\left(\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\right)}}d\kappa\bigg].
\end{align}
(33)
Setting \(\nu(v)=\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\) in 33), then by using (13) and assumption \(\mu(\nu(v))=\mu(\nu(\alpha)+m\nu(\beta)-m\nu(v))\), the first inequality of (32) is obtained.
Now multiplying (25) with \(h\left(\frac{1}{2}\right)\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p) {e^{\gamma\left(\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\right)}}\) and integrating over \([0,1]\), we have
\begin{align}\label{24}
&\notag h\left(\frac{1}{2}\right)\bigg[\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)e^{\mu\left( \frac{\kappa}{2}\nu(\alpha)+m\frac{(2-\kappa)}{2}\nu(\beta)\right)} {e^{\gamma\left(\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\right)}} d\kappa\\
\nonumber&+m\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p)e^{\mu\left(\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\right)} {e^{\gamma\left(\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\right)}}d\kappa\bigg]\\&\nonumber\leq\! h\!\left(\!\frac{1}{2}\!\right)\!\! \bigg[\!\!\left(\!e^{\mu(\nu(\alpha))}\!+\!me^{\mu(\nu(\beta))}\!\right)\!\!\int_{0}^{1}\!\!\!\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p) {e^{\gamma\left(\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\right)}}h\left(\frac{\kappa}{2}\right)\!d\kappa\\
&\;\;\;+\!m\!\!\left(\!e^{\mu(\nu(\beta))}\!+\!me^{\mu\left(\frac{\nu(\alpha)}{m^2}\!\right)}\right)\!\!\!\int_{0}^{1}\!\!\!\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p) {e^{\gamma\left(\!\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\!\right)}}h\!\left(\!\frac{2\!-\!\kappa}{2}\!\right)\!d\kappa\bigg].
\end{align}
(34)
Setting \(\nu(v)=\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}\frac{\nu(\alpha)}{m}\) in (34), then by using (13) and assumption \(\mu(\nu(v))=\mu(\nu(\alpha)+m\nu(\beta)-m\nu(v))\), the second inequality of (32) is obtained.
Corollary 4.
Setting \(m=1\) in (32), the following inequalities for exponentially \(h\)-convex function can be obtained:
\begin{align}\label{yy***}
&\notag{e^{\mu\left( \frac{\nu(\alpha)+\nu(\beta)}{2}\right)}}
\left(_{\nu}\Upsilon^{\varsigma,r,q,c}_{\sigma,\phi,l,\bar{\psi}2^\sigma,\left(\nu^{-1}\left(\frac{\nu(\alpha)+\nu(\beta)}{2}\right)\right) ^{-}}e^{\gamma\circ\nu}\right)\left(\alpha;p\right)\leq 2h\left(\frac{1}{2}\right) {\left(_{\nu}\Upsilon^{\varsigma,r,q,c}_{\sigma,\phi,l,\bar{\psi}2^\sigma,\left(\nu^{-1}\left(\frac{\nu(\alpha)+\nu(\beta)}{2}\right)\right) ^{-}}e^{\mu\circ\nu}e^{\gamma\circ\nu}\right)\left(\alpha;p\right)}\\\nonumber
&\leq h\left(\frac{1}{2}\right) \frac{(\nu(\beta)-\nu(\alpha))^\phi}{2^\phi}\left(e^{\mu(\nu(\alpha))}+e^{\mu(\nu(\beta))}\right)\bigg[\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p) {e^{\gamma\left(\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}{\nu(\alpha)}\right)}}h\left(\frac{\kappa}{2}\right)d\kappa\\
&\;\;\;+\int_{0}^{1}\kappa^{\phi-1}E_{\sigma,\phi,l}^{\varsigma,r,q,c}(\psi \kappa^{\sigma};p) {e^{\gamma\left(\frac{\kappa}{2}\nu(\beta)+\frac{(2-\kappa)}{2}{\nu(\alpha)}\right)}}h\left(\frac{2-\kappa}{2}\right)d\kappa\bigg],
\end{align}
(35)
where \(\bar{\psi}\) is same as in (21).
Remark 6.
- If we set \(h(\kappa)=\kappa\) in (32), then [33, Theorem 11] is obtained.
- If we set \(h(\kappa)=\kappa\) and \(m=1\) in (32), then [33, Corollary 4] is obtained.
Remark 7.
By setting \(h(\kappa)=\kappa^s\) and \(m=1\) in Theorems 12, 13, 14 and 15, the Hadamard and the Fejér-Hadamard inequalities for exponentially \(s\)-convex functions can be obtained. We leave it for interested reader.
4.Concluding remarks
In this article, we established the Hadamard and the Fejér-Hadamard inequalities. To established these inequalities generalized fractional integral operators and exponentially \((h,m)\)-convexity have been used. The presented results hold for various kind of exponentially convexity and well known fractional integral operators given in Remarks 1 and 2. Moreover, the established results have connection with already published results.
Acknowledgments
The research work of the Ghulam Farid is supported by Higher Education Commission of Pakistan under NRPU 2016, Project No. 5421.
Autho Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Conflict of Interests
The authors declare no conflict of interest.
