Fejér–Hadamard Type Inequalities for (α, h-m)-p-Convex Functions via Extended Generalized Fractional Integrals

Abstract

Integral operators of a fractional order containing the Mittag-Leffler function are important generalizations of classical Riemann–Liouville integrals. The inequalities that are extensively studied for fractional integral operators are the Hadamard type inequalities. The aim of this paper is to find new versions of the Fejér–Hadamard (weighted version of the Hadamard inequality) type inequalities for ($\alpha$, h-m)-p-convex functions via extended generalized fractional integrals containing Mittag-Leffler functions. These inequalities hold simultaneously for different types of well-known convexities as well as for different kinds of fractional integrals. Hence, the presented results provide more generalized forms of the Hadamard type inequalities as compared to the inequalities that already exist in the literature.

1. Introduction

Convexity in alliance with integral inequalities is an attractive area of research. Researchers define novel types of convexities for the need of hour. Convex functions and mathematical inequalities play a vital role in the progress of diverse fields of pure and applied sciences. A large number of inequalities have been established for convex functions, see [1,2,3,4,5,6,7,8,9,10] and references therein. On the other hand, fractional integral and derivative operators are important tools to generalize the classical concepts and methods that are based on ordinary integration and derivative. Fractional integral and derivative operators lead in the study of fractional differential equations [11], fractional initial and boundary value problems [12], and fractional dynamical systems [13].

The Mittag-Leffler function appears in the solutions of fractional differential equations, the same as how likely the exponential function appears in solving differential equations. Fractional integral operators containing Mittag-Leffler function are also developed and applied to study well-known real world problems, see [14,15,16,17,18,19] and references therein.

In recent years, fractional integral operators, as well as new classes of functions closely related to convex functions, play a significant role in extensions and generalizations of classical inequalities. The most celebrated inequality which is investigated for fractional integrals is the well-known Hadamard inequality. Sarikaya, in [6,20], proved two versions of the Hadamard inequality for convex functions by using Riemann–Liouville fractional integral operators. After that, plenty of such versions have been published by defining new kinds of functions related to convex functions via fractional integrals related to Riemann–Liouville integrals. Farid, in [21], gave the Hadamard and the Fejér–Hadamard inequalities for convex functions by using the fractional integral operators containing Mittag-Leffler functions. For other known and new classes of functions, the Hadamard and the Fejér–Hadamard fractional inequalities can be found in [22,23,24,25,26,27,28,29,30] and references therein.

Inspired by a huge number of findings in the credit of the Hadamard and the Fejér–Hadamard inequalities for fractional integrals, we are motivated in establishing the generalized forms of such type of inequalities by utilizing the class of so called ($\alpha$, h-m)-p-convex functions and the fractional integral operators involving Mittag-Leffler functions. Next, we give definitions and important notions which will be utilized in proving the results of this paper.

Definition 1

([31]). A function $f:[a,b]\to \mathbb{R}$ is said to be convex, if the following inequality holds:

$$f(ta+(1-t\left)b\right)\le tf\left(a\right)+(1-t)f\left(b\right),\forall t\in [0,1].$$

(1)

The Fejér–Hadamard inequality for convex functions is stated in the forthcoming theorem, while the Hadamard inequality can be deduced for $\hslash \equiv 1$.

Theorem 1

([32]). Let $f:[a,b]\to \mathbb{R}$ be a convex function with $a<b$. Then, the following inequality holds:

$$f\left(\frac{a+b}{2}\right){\int }_a^b\hslash \left(x\right)dx\le {\int }_a^{b}f\left(x\right)\hslash \left(x\right)dx\le \frac{f\left(a\right)+f\left(b\right)}{2}{\int }_a^b\hslash \left(x\right)dx,$$

where $\hslash :[a,b]\to \mathbb{R}$ is non-negative, integrable and symmetric function about $\frac{a+b}{2}$.

In [30], Jia et al. defined the class of ($\alpha$, h-m)-p-convex functions given as follows:

Definition 2.

Let $J\subseteq \mathbb{R}$ be an interval containing $(0,1)$. Let $I\subset (0,\infty )$ be a real interval and $p\in \mathbb{R}\backslash \left\{0\right\}$. Moreover, let $h:J\to \mathbb{R}$ be a non-negative function. A function $f:I\to \mathbb{R}$ is said to be (α, h-m)-p-convex, if

$$f\left({\left(t{a}^p+m(1-t)b^p\right)}^{\frac{1}{p}}\right)\le h\left(t^{\alpha }\right)f\left(a\right)+mh(1-t^{\alpha })f\left(b\right),\forall t\in (0,1),(\alpha ,m)\in {[0,1]}^2,$$

(2)

provided ${\left(t{a}^p+m(1-t)b^p\right)}^{\frac{1}{p}}\in I$.

The inequality (2), provides the definition of (h-m)-p-convex functions for $\alpha =1$; ($\alpha$,m)-p-convex functions for $h\left(t\right)=t$; ($\alpha$,h)-p-convex functions for $m=1$; (s,m)-p-convex functions for $h\left(t\right)=t^s,\alpha =1$; ($\alpha$, h-m)-convex functions for $p=1$; (h-m)-convex functions for $\alpha =p=1$; (h-p)-convex functions for $\alpha =m=1$; h-convex function for $\alpha =p=m=1$; s-convex functions for $h\left(t\right)=t^s, \alpha =p=m=1$; m-convex functions for $h\left(t\right)=t, \alpha =p=1$; p-convex functions for $h\left(t\right)=t, \alpha =m=1$; convex functions for $h\left(t\right)=t, \alpha =p=m=1$.

Next, we give the definition of integral operators involving Mittag-Leffler functions as follows:

Definition 3

([33]). Let $f,g:[a,b]\to \mathbb{R}$, $0<a<b$ be the functions such that f be positive and $f\in L_1[a,b]$ and g be a differentiable and strictly increasing. Moreover, let $\frac{\phi }{x}$ be an increasing function on $[a,\infty )$ and $\mu ,\psi ,\varphi ,\rho ,c\in \mathbb{C}$, $\Re \left(\psi \right), \Re \left(\varphi \right)>0$, $\Re \left(c\right)>\Re \left(\rho \right)>0$ with $q\ge 0$, $\sigma ,r>0$ and $0<k\le r+\sigma$. Then, for $x\in [a,b]$, the integral operators are defined by:

$$\begin{array}{c}\hfill \left({}_{g}F_{\sigma ,\psi ,\varphi ,\mu ,a^{+}}^{\phi ,\rho ,r,k,c}f\right)(x;q)={\int }_a^x\frac{\phi \left(g\right(x)-g(t\left)\right)}{g\left(x\right)-g\left(t\right)}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu {(g\left(x\right)-g\left(t\right))}^{\sigma };q)f\left(t\right)d\left(g\left(t\right)\right),\end{array}$$

(3)

$$\begin{array}{c}\hfill \left({}_{g}F_{\sigma ,\psi ,\varphi ,\mu ,b^{-}}^{\phi ,\rho ,r,k,c}f\right)(x;q)={\int }_x^b\frac{\phi \left(g\right(t)-g(x\left)\right)}{g\left(t\right)-g\left(x\right)}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu {(g\left(t\right)-g\left(x\right))}^{\sigma };q)f\left(t\right)d\left(g\left(t\right)\right).\end{array}$$

(4)

Here

$$E_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(t;q)=\sum _{n=0}^{\infty }\frac{{\beta }_q(\rho +nk,c-\rho ){\left(c\right)}_{nk}{t}^n}{\beta (\rho ,c-\rho )\Gamma (\sigma n+\psi ){\left(\varphi \right)}_{nr}},$$

(5)

is the generalized Mittag-Leffler function and ${\left(c\right)}_{nk}$ is the generalized Pochhammer symbol defined as follows:

$${\left(c\right)}_{nk}=\frac{\Gamma (c+nk)}{\Gamma \left(c\right)},$$

$\Gamma (.)$ is gamma function defined as follows:

$$\Gamma \left(\psi \right)={\int }_0^{\infty }{e}^{-t}{t}^{\psi -1}dt,\psi >0,$$

${\beta }_q$ is the extension of beta function defined as follows:

$${\beta }_q(x,y)={\int }_0^1{t}^{x-1}{(1-t)}^{y-1}{e}^{-\frac{q}{t(1-t)}}dt.$$

The following definition of extracted generalized fractional integral operators from Definition 3 is very useful to obtain the Fejér–Hadamard type inequalities.

Definition 4

([27]). Let $f,g:[a,b]\to \mathbb{R}$, $0<a<b$ be the functions such that f be positive and $f\in L_1[a,b]$ and g be a differentiable and strictly increasing. Moreover, let $\mu ,\psi ,\varphi ,\rho ,c\in \mathbb{C}$, $\Re \left(\psi \right),\Re \left(\varphi \right)>0$, $\Re \left(c\right)>\Re \left(\rho \right)>0$ with $q\ge 0$, $\sigma ,r>0$ and $0<k\le r+\sigma$. Then, for $x\in [a,b]$, the integral operators are defined by:

$$\begin{array}{c}\hfill \left({}_g{\rm Y}_{\sigma ,\psi ,\varphi ,\mu ,a^{+}}^{\rho ,r,k,c}f\right)(x;q)={\int }_a^x{(g\left(x\right)-g\left(t\right))}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu {(g\left(x\right)-g\left(t\right))}^{\sigma };q)f\left(t\right)d\left(g\left(t\right)\right),\end{array}$$

(6)

$$\begin{array}{c}\hfill \left({}_g{\rm Y}_{\sigma ,\psi ,\varphi ,\mu ,b^{-}}^{\rho ,r,k,c}f\right)(x;q)={\int }_x^b{(g\left(t\right)-g\left(x\right))}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu {(g\left(t\right)-g\left(x\right))}^{\sigma };q)f\left(t\right)d\left(g\left(t\right)\right).\end{array}$$

(7)

Remark 1.

The integral operators (6) and (7) provide the following fractional integral operators:

(i) 

For $g\left(x\right)=x$, we recover the fractional integral operators defined by Andrić et al. in [34].

(ii) 

For $g\left(x\right)=x$ and $q=0$, we recover the fractional integral operators defined by Salim-Faraj in [17].

(iii)

For $g\left(x\right)=x$ and $\varphi =r=1$, we recover the fractional integral operators defined by Rahman et al. in [16].

(iv)

For $g\left(x\right)=x$, $q=0$ and $\varphi =r=1$, we recover the fractional integral operators defined by Srivastava-Tomovski in [19].

(v)

For $g\left(x\right)=x$, $q=0$ and $\varphi =r=k=1$, we recover the fractional integral operators defined by Prabhakar in [15].

(vi)

For $g\left(x\right)=x$ and $\mu =q=0$, we recover the Riemann–Liouville fractional integrals.

In the next section, we give two versions of Fejér–Hadamard inequalities for $(\alpha ,h-m)-p$-convex functions via generalized fractional integral operators. Moreover, we give the Fejér–Hadamard inequalities for different classes of convex functions which are deducible from $(\alpha ,h-m)-p$-convex functions.

In the whole paper, we use the following notations frequently: $\left({\mathcal{F}}_{b,\sigma ,\psi }^{a^{+}}\right)(\mu ,f)=\left({}_g{{\rm Y}}_{\sigma ,\psi ,\varphi ,\mu ,a^{+}}^{\rho ,r,k,c}f\right)(b;q)$, $\left({\mathcal{F}}_{a,\sigma ,\psi }^{b^{-}}\right)(\mu ,f)=\left({}_g{{\rm Y}}_{\sigma ,\psi ,\varphi ,\mu ,b^{-}}^{\rho ,r,k,c}f\right)(a;q)$.

2. Fejér–Hadamard Type Inequalities

Theorem 2.

Let $f,g,\hslash :[a,b]\to \mathbb{R}$, $0<a<mb,m\in (0,1]$, Range $\left(g\right)$, Range $(\hslash )$$\subset [a,b]$ be the functions such that f is positive and $f\in L_1[a,b]$, g is differentiable and strictly increasing. If f is (α, h-m)-p-convex on $[a,b]$ and $\hslash \left({\left(g^p\left(a\right)+m{g}^p\left(b\right)-mg\left(y\right)\right)}^{\frac{1}{p}}\right)=\hslash \left({\left(g\left(y\right)\right)}^{\frac{1}{p}}\right)$, then for (6) and (7), we have the following inequalities:

(i) 

If $p>0$, then

$$\begin{array}{cc}\hfill & f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{+}}\right)({\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & \le h\left(\frac{1}{2^{\alpha }}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{+}}\right)({\mu }^{\prime },f\hslash \circ \zeta )+m^{\psi +1}h\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)\left({\mathcal{F}}_{g^{-1}\left(\frac{g^p\left(a\right)}{m}\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(b\right)\right)}^{-}}\right)(m^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(a\right)\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hfill \\ & \times \hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(t^{\alpha }\right)dt+m\left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(\frac{g\left(a\right)}{m^2}\right)\right)\hfill \\ & \times {\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(1-t^{\alpha }\right)dt,\hfill \end{array}$$

(8)

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [a^p,m{b}^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(m{g}^p\left(b\right)-g^p\left(a\right))}^{\sigma }}$.

(ii) 

If $p<0$, then

$$\begin{array}{cc}\hfill & f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{-}}\right)({\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & \le h\left(\frac{1}{2^{\alpha }}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{-}}\right)({\mu }^{\prime },f\hslash \circ \zeta )+m^{\psi +1}h\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)\left({\mathcal{F}}_{g^{-1}\left(\frac{g^p\left(a\right)}{m}\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(b\right)\right)}^{+}}\right)(m^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(a\right)\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hfill \\ & \times \hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(t^{\alpha }\right)dt+m\left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(\frac{g\left(a\right)}{m^2}\right)\right)\hfill \\ & \times {\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(1-t^{\alpha }\right)dt,\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [m{b}^p,a^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(g^p\left(a\right)-m{g}^p\left(b\right))}^{\sigma }}$.

Proof. 

(i) As we know, f is ($\alpha$, h-m)-p-convex function; therefore, we can write

$$\begin{array}{cc}\hfill & f\left({\left(\frac{g^p\left(x\right)+m{g}^p\left(y\right)}{2}\right)}^{\frac{1}{p}}\right)\le h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(x\right)\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(g\left(y\right)\right).\hfill \end{array}$$

(9)

Putting $g\left(x\right)={\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}$ and $g\left(y\right)={\left(t{g}^p\left(b\right)+(1-t)\frac{g^p\left(a\right)}{m}\right)}^{\frac{1}{p}}$ in (9), we obtain

$$\begin{array}{c}f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\hfill \\ \le h\left(\frac{1}{2^{\alpha }}\right)f\left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left({\left(t{g}^p\left(b\right)+(1-t)\frac{g^p\left(a\right)}{m}\right)}^{\frac{1}{p}}\right).\hfill \end{array}$$

(10)

Multiplying (10) by $t^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)$ and integrating, we have

$$\begin{array}{cc}\hfill & f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)dt\hfill \\ & \le h\left(\frac{1}{2^{\alpha }}\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)f\left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)\hfill \\ & \times \hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)dt+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hfill \\ & f\left({\left(t{g}^p\left(b\right)+(1-t)\frac{g^p\left(a\right)}{m}\right)}^{\frac{1}{p}}\right)\hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)dt.\hfill \end{array}$$

We use substitution $t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)=g\left(x\right)$ in the integral appearing on the left hand side. In the integrals appearing on the right hand side, we use substitution $g\left(y\right)=\left(t{g}^p\left(b\right)+(1-t)\frac{g^p\left(a\right)}{m}\right)$ that is $t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)=g^p\left(a\right)+m{g}^p\left(b\right)-mg\left(y\right)$. By utilizing the condition $\hslash \left({\left(g^p\left(a\right)+m{g}^p\left(b\right)-mg\left(y\right)\right)}^{\frac{1}{p}}\right)=\hslash \left({\left(g\left(y\right)\right)}^{\frac{1}{p}}\right)$ and the Equations (6) and (7), the first inequality of (8) can be achieved.

From the right hand side of (10) for ($\alpha$, h-m)-p-convex function f, we can write

$$\begin{array}{cc}\hfill & h\left(\frac{1}{2^{\alpha }}\right)f\left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left({\left(t{g}^p\left(b\right)+(1-t)\frac{g^p\left(a\right)}{m}\right)}^{\frac{1}{p}}\right)\hfill \\ \hfill & \le \left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(a\right)\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)\right)h\left(t^{\alpha }\right)\hfill \\ & +m\left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(\frac{g\left(a\right)}{m^2}\right)\right)h\left(1-t^{\alpha }\right).\hfill \end{array}$$

(11)

Now, multiplying (11) by $t^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)$ and integrating, we have

$$\begin{array}{cc}\hfill & h\left(\frac{1}{2^{\alpha }}\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)f\left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)\hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)dt\hfill \\ & +mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)f\left({\left(t{g}^p\left(b\right)+(1-t)\frac{g^p\left(a\right)}{m}\right)}^{\frac{1}{p}}\right)\hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)dt\hfill \\ & \le \left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(a\right)\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(t^{\alpha }\right)dt\hfill \\ & +m\left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(\frac{g\left(a\right)}{m^2}\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hfill \\ & \times \hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(1-t^{\alpha }\right)dt.\hfill \end{array}$$

Again, setting $t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)=g\left(x\right)$ in the first term of left hand side. While using $g\left(y\right)=\left(t{g}^p\left(b\right)+(1-t)\frac{g^p\left(a\right)}{m}\right)$, that is $t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)=g^p\left(a\right)+m{g}^p\left(b\right)-mg\left(y\right)$ in the second term of left hand side and utilizing condition $\hslash \left({\left(g^p\left(a\right)+m{g}^p\left(b\right)-mg\left(y\right)\right)}^{\frac{1}{p}}\right)=\hslash \left({\left(g\left(y\right)\right)}^{\frac{1}{p}}\right)$. Then, using the Equations (6) and (7), the second inequality of (8) is achieved.

(ii) Proof is similar to the proof of (i). □

Remark 2.

(i) For $g=I$ and $\mu =q=0$ in Theorem 2 (i), Theorem 3.1 [30] is achieved.

(ii) For $\alpha =m=p=1$ and $h\left(t\right)=t$ in Theorem 2 (i), Theorem 7 [27] is achieved.

(iii) For $\alpha =m=1$, $p=-1$ and $h\left(t\right)=t$ in Theorem 2 (ii), Theorem 2.5 [26] is achieved.

(iv) For $\alpha =m=1$, $p=-1$, $\mu =q=0$, $g=I$, $\hslash \left(x\right)=1$ and $h\left(t\right)=t$ in Theorem 2 (ii), Theorem 4 [24] is achieved.

(v) For $\alpha =m=1$, $p=-1$, $\mu =q=0$, $g=I$, $\psi =1$ and $h\left(t\right)=t$ in Theorem 2 (ii), Theorem 8 [3] is achieved.

(vi) For $\alpha =m=1$, $p=-1$, $\mu =q=0$, $g=I$, $\psi =1$, $\hslash \left(x\right)=1$ and $h\left(t\right)=t$ in Theorem 2 (ii), Theorem 2.4 [5] is achieved.

(vii) For $\alpha =m=1$, $p>0$ and $h\left(t\right)=t$ in Theorem 2 (i), Theorem 4 (i) [29] is achieved.

(viii) For $\alpha =m=1$, $p<0$ and $h\left(t\right)=t$ in Theorem 2 (ii), Theorem 4 (ii) [29] is achieved.

(ix) For $p=-1$ in Theorem 2 (ii), Theorem 4 [25] is achieved.

(x) For $\alpha =m=1$, $p=1$, $h\left(t\right)=t$, $\mu =q=0$, $g=I$ and $\psi =1$ in Theorem 2 (i), classical Fejér–Hadamard inequality [32] is achieved.

(xi) For $\alpha =m=1$, $p=1$, $h\left(t\right)=t$, $\mu =q=0$, $g=I$, $\hslash \left(x\right)=1$ and $\psi =1$ in Theorem 2 (i), classical Hadamard inequality [35,36] is achieved.

(xii) For $p=1$ in Theorem 2 (i), the result for (α, h-m)-convex function is achieved.

(xiii) For $\alpha =1$, $p=1$, in Theorem 2 (i), Theorem 2.2 [28] is achieved. Further, if $h\left(t\right)=t$, then Theorem 2.1 [28] is achieved.

(xiv) For $p=1$ and $h\left(t\right)=t$ in Theorem 2 (i), the result for (α, m)-convex function is achieved.

(xv) For $\alpha =1$ and $p=1$ in Theorem 2 (i), the result for (h-m)-convex function is achieved.

Theorem 3.

Under the suppositions of Theorem 2, we have the following inequalities:

(i) 

If $p>0$, then

$$\begin{array}{cc}\hfill & f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{+}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )\le h\left(\frac{1}{2^{\alpha }}\right)\hfill \\ & \times \left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{+}}\right)(2^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )+m^{\psi +1}h\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)\left({\mathcal{F}}_{g^{-1}\left(\frac{g^p\left(a\right)}{m}\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2m}\right)}^{-}}\right)({\left(2m\right)}^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ & \le \left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(a\right)\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hfill \\ & \times \hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left({\left(\frac{t}{2}\right)}^{\alpha }\right)dt+m\left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(\frac{g\left(a\right)}{m^2}\right)\right)\hfill \\ & \times {\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(1-{\left(\frac{t}{2}\right)}^{\alpha }\right)dt,\hfill \end{array}$$

(12)

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [a^p,m{b}^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(m{g}^p\left(b\right)-g^p\left(a\right))}^{\sigma }}$.

(ii) 

If $p<0$, then

$$\begin{array}{cc}\hfill & f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{-}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )\le h\left(\frac{1}{2^{\alpha }}\right)\hfill \\ & \times \left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{-}}\right)(2^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )+m^{\psi +1}h\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)\left({\mathcal{F}}_{g^{-1}\left(\frac{g^p\left(a\right)}{m}\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2m}\right)}^{+}}\right)({\left(2m\right)}^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(a\right)\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hfill \\ & \times \hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left({\left(\frac{t}{2}\right)}^{\alpha }\right)dt+m\left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(\frac{g\left(a\right)}{m^2}\right)\right)\hfill \\ & \times {\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(1-{\left(\frac{t}{2}\right)}^{\alpha }\right)dt,\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [m{b}^p,a^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(g^p\left(a\right)-m{g}^p\left(b\right))}^{\sigma }}$.

Proof. 

(i) Putting $g\left(x\right)={\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}$ and $g\left(y\right)={\left(\frac{t}{2}{g}^p\left(b\right)+\left(1-\frac{t}{2}\right)\frac{g^p\left(a\right)}{m}\right)}^{\frac{1}{p}}$ in (9), we obtain

$$\begin{array}{cc}\hfill & f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\hfill \\ & \le h\left(\frac{1}{2^{\alpha }}\right)f\left({\left(\frac{t}{2}{g}^p\left(a\right)+m(1-\frac{t}{2})g^p\left(b\right)\right)}^{\frac{1}{p}}\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left({\left(\frac{t}{2}{g}^p\left(b\right)+(1-\frac{t}{2})\frac{g^p\left(a\right)}{m}\right)}^{\frac{1}{p}}\right).\hfill \end{array}$$

(13)

Multiplying (13) by $t^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)$ and integrating, we have

$$\begin{array}{cc}\hfill & f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)dt\hfill \\ & \le h\left(\frac{1}{2^{\alpha }}\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)f\left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)\hfill \\ & \times \hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)dt+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hfill \\ & f\left({\left(\frac{t}{2}{g}^p\left(b\right)+\left(1-\frac{t}{2}\right)\frac{g^p\left(a\right)}{m}\right)}^{\frac{1}{p}}\right)\hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)dt.\hfill \end{array}$$

We use the substitution $\frac{t}{2}{g}^p\left(a\right)+m(1-\frac{t}{2})g^p\left(b\right)=g\left(x\right)$ in integral appearing in the left hand side. While in the integral appearing in the first term of the right hand side, we use $g\left(y\right)=\left(\frac{t}{2}{g}^p\left(b\right)+(1-\frac{t}{2})\frac{g^p\left(a\right)}{m}\right)$ that is $\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)=g^p\left(a\right)+m{g}^p\left(b\right)-mg\left(y\right)$ in the last term of right hand side. Then, by utilizing the given condition $\hslash \left({\left(g^p\left(a\right)+m{g}^p\left(b\right)-mg\left(y\right)\right)}^{\frac{1}{p}}\right)=\hslash \left({\left(g\left(y\right)\right)}^{\frac{1}{p}}\right)$ and the Equations (6) and (7), the first inequality of (12) can be achieved.

From the right hand side of (13) for ($\alpha$, h-m)-p-convex function f, we can write

$$\begin{array}{cc}\hfill & h\left(\frac{1}{2^{\alpha }}\right)f\left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left({\left(\frac{t}{2}{g}^p\left(b\right)+\left(1-\frac{t}{2}\right)\frac{g^p\left(a\right)}{m}\right)}^{\frac{1}{p}}\right)\hfill \\ & \le \left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(a\right)\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)\right)h\left({\left(\frac{t}{2}\right)}^{\alpha }\right)\hfill \\ & +m\left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(\frac{g\left(a\right)}{m^2}\right)\right)h\left(1-{\left(\frac{t}{2}\right)}^{\alpha }\right).\hfill \end{array}$$

(14)

Now, multiplying (14) by $t^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)$ and integrating over $[0,1]$, we have

$$\begin{array}{cc}\hfill & h\left(\frac{1}{2^{\alpha }}\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)f\left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)\hfill \\ & \times \hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)dt+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hfill \\ & f\left({\left(\frac{t}{2}{g}^p\left(b\right)+\left(1-\frac{t}{2}\right)\frac{g^p\left(a\right)}{m}\right)}^{\frac{1}{p}}\right)\hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)dt\hfill \\ & \le \left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(a\right)\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hfill \\ & \times \hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left({\left(\frac{t}{2}\right)}^{\alpha }\right)dt+m\left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)+mh\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)\right.\hfill \\ & \left.f\left(\frac{g\left(a\right)}{m^2}\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(1-{\left(\frac{t}{2}\right)}^{\alpha }\right)dt.\hfill \end{array}$$

Again, we set $\frac{t}{2}{g}^p\left(a\right)+m(1-\frac{t}{2})g^p\left(b\right)=g\left(x\right)$ in integral appearing in the first term of the left hand side. While we use $g\left(y\right)=\left(\frac{t}{2}{g}^p\left(b\right)+(1-\frac{t}{2})\frac{g^p\left(a\right)}{m}\right)$ that is $\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)=g^p\left(a\right)+m{g}^p\left(b\right)-mg\left(y\right)$ in integral appearing in the second term of the left hand side. By utilizing the condition $\hslash \left({\left(g^p\left(a\right)+m{g}^p\left(b\right)-mg\left(y\right)\right)}^{\frac{1}{p}}\right)=\hslash \left({\left(g\left(y\right)\right)}^{\frac{1}{p}}\right)$ and the Equations (6) and (7), the second inequality of (12) can be achieved.

(ii) Proof is similar to the proof of (i). □

Remark 3.

(i) For $g=I$ and $\mu =q=0$ in Theorem 3 (i), Theorem 3.3 [30] is achieved.

(ii) For $\alpha =m=1$, $p=-1$ and $h\left(t\right)=t$ in Theorem 3 (ii), Theorem 3.3 [26] is achieved.

(iii) For $\alpha =m=1$, $p=-1$ and $h\left(t\right)=t$ in Theorem 3 (ii), Theorem 8 [3] is achieved.

(iv) For $\alpha =m=1$, $p=-1$, $\mu =q=0$, $g=I$ and $h\left(t\right)=t$ in Theorem 3 (ii), Corollary 2.10 [26] is achieved.

(v) For $\alpha =m=1$, $p>0$ and $h\left(t\right)=t$ in Theorem 3 (i), Theorem 6 (i) [29] is achieved.

(vi) For $\alpha =m=1$, $p<0$ and $h\left(t\right)=t$ in Theorem 3 (ii), Theorem 6 (ii) [29] is achieved.

(vii) For $p=-1$ in Theorem 3 (ii), Theorem 6 [25] is achieved.

(viii) For $p=1$ in Theorem 3 (i), the result for (α, h-m)-convex function is achieved.

(ix) For $p=1$ and $h\left(t\right)=t$ in Theorem 3 (i), the result for (α, m)-convex function is achieved.

(x) For $\alpha =1$ and $p=1$ in Theorem 3 (i), the result for (h-m)-convex function is achieved.

In the following, we list the results for (h-m)-p-convex, ($\alpha$,m)-p-convex, ($\alpha$,h)-p-convex and (s,m)-p-convex functions.

2.1. Results for (h-m)-p-Convex Functions

Theorem 4.

From Theorem 2 for $\alpha =1$, we have the following inequalities for (h-m)-p-convex functions:

(i) 

If $p>0$, then

$$\begin{array}{cc}\hfill & \frac{1}{h\left(\frac{1}{2}\right)}f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{+}}\right)({\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & \le \left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{+}}\right)({\mu }^{\prime },f\hslash \circ \zeta )+m^{\psi +1}\left({\mathcal{F}}_{g^{-1}\left(\frac{g^p\left(a\right)}{m}\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(b\right)\right)}^{-}}\right)(m^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \left(f\left(g\right(a\left)\right)+mf\left(g\right(b\left)\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(t\right)dt\hfill \\ & +m\left(f\left(g\left(b\right)\right)+mf\left(\frac{g\left(a\right)}{m^2}\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(1-t\right)dt,\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [a^p,m{b}^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(m{g}^p\left(b\right)-g^p\left(a\right))}^{\sigma }}$.

(ii) 

If $p<0$, then

$$\begin{array}{cc}\hfill & \frac{1}{h\left(\frac{1}{2}\right)}f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{-}}\right)({\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & \le \left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{=}}\right)({\mu }^{\prime },f\hslash \circ \zeta )+m^{\psi +1}\left({\mathcal{F}}_{g^{-1}\left(\frac{g^p\left(a\right)}{m}\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(b\right)\right)}^{+}}\right)(m^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \left(f\left(g\right(a\left)\right)+mf\left(g\right(b\left)\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(t\right)dt\hfill \\ & +m\left(f\left(g\left(b\right)\right)+mf\left(\frac{g\left(a\right)}{m^2}\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(1-t\right)dt,\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [m{b}^p,a^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(g^p\left(a\right)-m{g}^p\left(b\right))}^{\sigma }}$.

Theorem 5.

From Theorem 3, for $\alpha =1$, we have the following inequalities for (h-m)-p-convex functions:

(i) 

If $p>0$, then

$$\begin{array}{cc}\hfill & \frac{1}{h\left(\frac{1}{2}\right)}f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{+}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & \le \left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{+}}\right)(2^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )+m^{\psi +1}\left({\mathcal{F}}_{g^{-1}\left(\frac{g^p\left(a\right)}{m}\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2m}\right)}^{-}}\right)({\left(2m\right)}^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \left(f\left(g\right(a\left)\right)+mf\left(g\right(b\left)\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(\frac{t}{2}\right)dt\hfill \\ & +m\left(f\left(g\left(b\right)\right)+mf\left(\frac{g\left(a\right)}{m^2}\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(1-\frac{t}{2}\right)dt,\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [a^p,m{b}^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(m{g}^p\left(b\right)-g^p\left(a\right))}^{\sigma }}$.

(ii) 

If $p<0$, then

$$\begin{array}{cc}\hfill & \frac{1}{h\left(\frac{1}{2}\right)}f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{-}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & \le \left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{-}}\right)(2^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )+m^{\psi +1}\left({\mathcal{F}}_{g^{-1}\left(\frac{g^p\left(a\right)}{m}\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2m}\right)}^{+}}\right)({\left(2m\right)}^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \left(f\left(g\right(a\left)\right)+mf\left(g\right(b\left)\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(\frac{t}{2}\right)dt\hfill \\ & +m\left(f\left(g\left(b\right)\right)+mf\left(\frac{g\left(a\right)}{m^2}\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(1-\frac{t}{2}\right)dt,\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [m{b}^p,a^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(g^p\left(a\right)-m{g}^p\left(b\right))}^{\sigma }}$.

2.2. Results for ($\alpha$,m)-p-Convex Functions

Theorem 6.

From Theorem 2 for $h\left(t\right)=t$, we have the following inequalities for (α,m)-p-convex functions:

(i) 

If $p>0$, then

$$\begin{array}{cc}\hfill & 2^{\alpha }f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{+}}\right)({\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & \le \left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{+}}\right)({\mu }^{\prime },f\hslash \circ \zeta )+m^{\psi +1}\left(2^{\alpha }-1\right)\left({\mathcal{F}}_{g^{-1}\left(\frac{g^p\left(a\right)}{m}\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(b\right)\right)}^{-}}\right)(m^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \left(f\left(g\left(a\right)\right)+m\left(2^{\alpha }-1\right)f\left(g\left(b\right)\right)\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi +\alpha }^{{g^{-1}\left(g^p\left(a\right)\right)}^{+}}\right)({\mu }^{\prime },\hslash \circ \zeta )+m\left(f\left(g\left(b\right)\right)+m\left(2^{\alpha }-1\right)\right.\hfill \\ & \left.f\left(\frac{g\left(a\right)}{m^2}\right)\right)\left(\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{+}}\right)({\mu }^{\prime },\hslash \circ \zeta )-\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi +\alpha }^{{g^{-1}\left(g^p\left(a\right)\right)}^{+}}\right)({\mu }^{\prime },\hslash \circ \zeta )\right),\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [a^p,m{b}^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(m{g}^p\left(b\right)-g^p\left(a\right))}^{\sigma }}$.

(ii) 

If $p<0$, then

$$\begin{array}{cc}\hfill & 2^{\alpha }f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{-}}\right)({\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & \le \left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{-}}\right)({\mu }^{\prime },f\hslash \circ \zeta )+m^{\psi +1}\left(2^{\alpha }-1\right)\left({\mathcal{F}}_{g^{-1}\left(\frac{g^p\left(a\right)}{m}\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(b\right)\right)}^{+}}\right)(m^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \left(f\left(g\left(a\right)\right)+m\left(2^{\alpha }-1\right)f\left(g\left(b\right)\right)\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi +\alpha }^{{g^{-1}\left(g^p\left(a\right)\right)}^{-}}\right)({\mu }^{\prime },\hslash \circ \zeta )+m\left(f\left(g\left(b\right)\right)+m\left(2^{\alpha }-1\right)\right.\hfill \\ & \left.f\left(\frac{g\left(a\right)}{m^2}\right)\right)\left(\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{-}}\right)({\mu }^{\prime },\hslash \circ \zeta )-\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi +\alpha }^{{g^{-1}\left(g^p\left(a\right)\right)}^{-}}\right)({\mu }^{\prime },\hslash \circ \zeta )\right),\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [m{b}^p,a^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(g^p\left(a\right)-m{g}^p\left(b\right))}^{\sigma }}$.

Theorem 7.

From Theorem 3 for $h\left(t\right)=t$, we have the following inequalities for (α,m)-p-convex functions:

(i) 

If $p>0$, then

$$\begin{array}{cc}\hfill & 2^{\alpha }f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{+}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & \le \left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{+}}\right)(2^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )+m^{\psi +1}\left(2^{\alpha }-1\right)\left({\mathcal{F}}_{g^{-1}\left(\frac{g^p\left(a\right)}{m}\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2m}\right)}^{-}}\right)({\left(2m\right)}^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \frac{1}{2^{\alpha }}\left(f\left(g\left(a\right)\right)+m\left(2^{\alpha }-1\right)f\left(g\left(b\right)\right)\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi +\alpha }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{+}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )+m\left(f\left(g\left(b\right)\right)+m\left(2^{\alpha }-1\right)\right.\hfill \\ & \left.f\left(\frac{g\left(a\right)}{m^2}\right)\right)\left(\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{+}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )-\frac{1}{2^{\alpha }}\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi +\alpha }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{+}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )\right),\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [a^p,m{b}^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(m{g}^p\left(b\right)-g^p\left(a\right))}^{\sigma }}$.

(ii) 

If $p<0$, then

$$\begin{array}{cc}\hfill & 2^{\alpha }f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{-}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & \le \left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{-}}\right)(2^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )+m^{\psi +1}\left(2^{\alpha }-1\right)\left({\mathcal{F}}_{g^{-1}\left(\frac{g^p\left(a\right)}{m}\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2m}\right)}^{+}}\right)({\left(2m\right)}^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \frac{1}{2^{\alpha }}\left(f\left(g\left(a\right)\right)+m\left(2^{\alpha }-1\right)f\left(g\left(b\right)\right)\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi +\alpha }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{-}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )+m\left(f\left(g\left(b\right)\right)+m\left(2^{\alpha }-1\right)\right.\hfill \\ & \left.f\left(\frac{g\left(a\right)}{m^2}\right)\right)\left(\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{-}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )-\frac{1}{2^{\alpha }}\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi +\alpha }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{-}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )\right),\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [m{b}^p,a^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(g^p\left(a\right)-m{g}^p\left(b\right))}^{\sigma }}$.

2.3. Results for ($\alpha$,h)-p-Convex Functions

Theorem 8.

From Theorem 2 for $m=1$, we have the following inequalities for (α,h)-p-convex functions:

(i) 

If $p>0$, then

$$\begin{array}{cc}\hfill & f\left({\left(\frac{g^p\left(a\right)+g^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(g^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{+}}\right)({\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & \le h\left(\frac{1}{2^{\alpha }}\right)\left({\mathcal{F}}_{g^{-1}\left(g^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{+}}\right)({\mu }^{\prime },f\hslash \circ \zeta )+h\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)\left({\mathcal{F}}_{g^{-1}\left(g^p\left(a\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(b\right)\right)}^{-}}\right)({\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(a\right)\right)+h\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hfill \\ & \times \hslash \left({\left(t{g}^p\left(a\right)+(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(t^{\alpha }\right)dt+\left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)+h\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(g\left(a\right)\right)\right)\hfill \\ & \times {\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(t{g}^p\left(a\right)+(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(1-t^{\alpha }\right)dt,\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right)$, $z\in [a^p,b^p]$, $f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta )$, ${\mu }^{\prime }=\frac{\mu }{{(g^p\left(b\right)-g^p\left(a\right))}^{\sigma }}$.

(ii) 

If $p<0$, then

$$\begin{array}{cc}\hfill & f\left({\left(\frac{g^p\left(a\right)+g^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(g^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{-}}\right)({\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & \le h\left(\frac{1}{2^{\alpha }}\right)\left({\mathcal{F}}_{g^{-1}\left(g^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{-}}\right)({\mu }^{\prime },f\hslash \circ \zeta )+h\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)\left({\mathcal{F}}_{g^{-1}\left(g^p\left(a\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(b\right)\right)}^{+}}\right)({\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(a\right)\right)+h\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hfill \\ & \times \hslash \left({\left(t{g}^p\left(a\right)+(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(t^{\alpha }\right)dt+\left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)+h\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(g\left(a\right)\right)\right)\hfill \\ & \times {\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(t{g}^p\left(a\right)+(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(1-t^{\alpha }\right)dt,\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right)$, $z\in [b^p,a^p]$, ${\mu }^{\prime }=\frac{\mu }{{(g^p\left(a\right)-g^p\left(b\right))}^{\sigma }}$.

Theorem 9.

From Theorem 3 for $m=1$, we have the following inequalities for (α,h)-p-convex functions:

(i) 

If $p>0$, then

$$\begin{array}{cc}\hfill & f\left({\left(\frac{g^p\left(a\right)+g^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(g^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+g^p\left(b\right)}{2}\right)}^{+}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )\le h\left(\frac{1}{2^{\alpha }}\right)\hfill \\ & \times \left({\mathcal{F}}_{g^{-1}\left(g^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+g^p\left(b\right)}{2}\right)}^{+}}\right)(2^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )+h\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)\left({\mathcal{F}}_{g^{-1}\left(g^p\left(a\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+g^p\left(b\right)}{2}\right)}^{-}}\right)(2^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(a\right)\right)+h\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hfill \\ & \times \hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left({\left(\frac{t}{2}\right)}^{\alpha }\right)dt+\left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)+h\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(g\left(a\right)\right)\right)\hfill \\ & \times {\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(1-{\left(\frac{t}{2}\right)}^{\alpha }\right)dt,\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [a^p,b^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(g^p\left(b\right)-g^p\left(a\right))}^{\sigma }}$.

(ii) 

If $p<0$, then

$$\begin{array}{cc}\hfill & f\left({\left(\frac{g^p\left(a\right)+g^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(g^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+g^p\left(b\right)}{2}\right)}^{-}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )\le h\left(\frac{1}{2^{\alpha }}\right)\hfill \\ & \times \left({\mathcal{F}}_{g^{-1}\left(g^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+g^p\left(b\right)}{2}\right)}^{-}}\right)(2^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )+h\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)\left({\mathcal{F}}_{g^{-1}\left(g^p\left(a\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+g^p\left(b\right)}{2}\right)}^{+}}\right)(2^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(a\right)\right)+h\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)\right){\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hfill \\ & \times \hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left({\left(\frac{t}{2}\right)}^{\alpha }\right)dt+\left(h\left(\frac{1}{2^{\alpha }}\right)f\left(g\left(b\right)\right)+h\left(\frac{2^{\alpha }-1}{2^{\alpha }}\right)f\left(g\left(a\right)\right)\right)\hfill \\ & \times {\int }_0^1{t}^{\psi -1}{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)h\left(1-{\left(\frac{t}{2}\right)}^{\alpha }\right)dt,\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [b^p,a^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(g^p\left(a\right)-g^p\left(b\right))}^{\sigma }}$.

2.4. Results for (s-m)-p-Convex Functions

Theorem 10.

From Theorem 2 for $\alpha =1$ and $h\left(t\right)=t^s$, we have the following inequalities for (s-m)-p-convex functions:

(i) 

If $p>0$, then

$$\begin{array}{cc}\hfill & 2^{s}f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{+}}\right)({\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & \le \left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{+}}\right)({\mu }^{\prime },f\hslash \circ \zeta )+m^{\psi +1}\left({\mathcal{F}}_{g^{-1}\left(\frac{g^p\left(a\right)}{m}\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(b\right)\right)}^{-}}\right)(m^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \left(f\left(g\right(a\left)\right)+mf\left(g\right(b\left)\right)\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi +s}^{{g^{-1}\left(g^p\left(a\right)\right)}^{+}}\right)({\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & +{\int }_0^1{t}^{\psi -1}{(1-t)}^s{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)dt,\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [a^p,m{b}^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(m{g}^p\left(b\right)-g^p\left(a\right))}^{\sigma }}$.

(ii) 

If $p<0$, then

$$\begin{array}{cc}\hfill & 2^{s}f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{-}}\right)({\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & \le \left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(a\right)\right)}^{+}}\right)({\mu }^{\prime },f\hslash \circ \zeta )+m^{\psi +1}\left({\mathcal{F}}_{g^{-1}\left(\frac{g^p\left(a\right)}{m}\right),\sigma ,\psi }^{{g^{-1}\left(g^p\left(b\right)\right)}^{-}}\right)(m^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \left(f\left(g\right(a\left)\right)+mf\left(g\right(b\left)\right)\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi +s}^{{g^{-1}\left(g^p\left(a\right)\right)}^{-}}\right)({\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & +{\int }_0^1{t}^{\psi -1}{(1-t)}^s{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(t{g}^p\left(a\right)+m(1-t)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)dt,\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [m{b}^p,a^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(g^p\left(a\right)-m{g}^p\left(b\right))}^{\sigma }}$.

Theorem 11.

From Theorem 3 for $\alpha =1$ and $h\left(t\right)=t^s$, we have the following inequalities for (s-m)-p-convex functions:

(i) 

If $p>0$, then

$$\begin{array}{cc}\hfill & 2^{s}f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{+}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & \le \left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{+}}\right)(2^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )+m^{\psi +1}\left({\mathcal{F}}_{g^{-1}\left(\frac{g^p\left(a\right)}{m}\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2m}\right)}^{+}}\right)({\left(2m\right)}^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \frac{1}{2^s}\left(f\left(g\right(a\left)\right)+mf\left(g\right(b\left)\right)\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi +s}^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{+}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & +m\left(f\left(g\left(b\right)\right)+mf\left(\frac{g\left(a\right)}{m^2}\right)\right){\int }_0^1{t}^{\psi -1}{\left(1-\frac{t}{2}\right)}^s{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)dt,\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [a^p,m{b}^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(m{g}^p\left(b\right)-g^p\left(a\right))}^{\sigma }}$.

(ii) 

If $p<0$, then

$$\begin{array}{cc}\hfill & 2^{s}f\left({\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{\frac{1}{p}}\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{-}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & \le \left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{-}}\right)(2^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )+m^{\psi +1}\left({\mathcal{F}}_{g^{-1}\left(\frac{g^p\left(a\right)}{m}\right),\sigma ,\psi }^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2m}\right)}^{+}}\right)({\left(2m\right)}^{\sigma }{\mu }^{\prime },f\hslash \circ \zeta )\hfill \\ \hfill & \le \frac{1}{2^s}\left(f\left(g\right(a\left)\right)+mf\left(g\right(b\left)\right)\right)\left({\mathcal{F}}_{g^{-1}\left(m{g}^p\left(b\right)\right),\sigma ,\psi +s}^{{g^{-1}\left(\frac{g^p\left(a\right)+m{g}^p\left(b\right)}{2}\right)}^{-}}\right)(2^{\sigma }{\mu }^{\prime },\hslash \circ \zeta )\hfill \\ & +m\left(f\left(g\left(b\right)\right)+mf\left(\frac{g\left(a\right)}{m^2}\right)\right){\int }_0^1{t}^{\psi -1}{\left(1-\frac{t}{2}\right)}^s{E}_{\sigma ,\psi ,\varphi }^{\rho ,r,k,c}(\mu t^{\sigma };q)\hslash \left({\left(\frac{t}{2}{g}^p\left(a\right)+m\left(1-\frac{t}{2}\right)g^p\left(b\right)\right)}^{\frac{1}{p}}\right)dt,\hfill \end{array}$$

$\zeta \left(z\right)=g^{\frac{1}{p}}\left(z\right),z\in [m{b}^p,a^p],f\hslash \circ \zeta =(f\circ \zeta )(\hslash \circ \zeta ),{\mu }^{\prime }=\frac{\mu }{{(g^p\left(a\right)-m{g}^p\left(b\right))}^{\sigma }}$.

Remark 4.

From Theorems 2 and 3, one can deduce results for (s,m)$-p$-Godunova–Levin-convex function of second kind, (p,P)-convex function, Godunova–Levin type harmonic convex function, s-Godunova–Levin type harmonic convex function, (α,h-m)-HA-convex function, (α,h)-HA-convex function, HA-convex function and (α,m)-HA-convex function. Moreover, all the results for operators given in Remark 1 [27] can be obtained.

3. Conclusions

It is common practice to establish the Hadamard type integral inequalities for new classes of functions related to convex functions. On the other hand, fractional integral operators are used to provide the generalizations of these inequalities. This paper presents the inequalities of Fejér–Hadamard type which simultaneously hold for many kinds of fractional integral operators. The reader can deduce a number of published as well as new Hadamard and Fejér–Hadamard type inequalities from the results of this paper.