Fractional Integral Inequalities of Hermite–Hadamard Type for ( h , g ; m ) -Convex Functions with Extended Mittag-Lefﬂer Function

: Several fractional integral inequalities of the Hermite–Hadamard type are presented for the class of ( h , g ; m ) -convex functions. Applied fractional integral operators contain extended generalized Mittag-Lefﬂer functions as their kernel, thus enabling new fractional integral inequalities that extend and generalize the known results. As an application, the upper bounds of fractional integral operators for ( h , g ; m ) -convex functions are given.


Introduction
In recent years, in the field of applied sciences, fractional calculus has been used with different boundary conditions to develop mathematical models relating to real-world problems. This significant interest in the theory of fractional calculus has been stimulated by many of its applications, especially in the various fields of physics and engineering.
Inequalities involving integrals of functions and their derivatives are of great importance in mathematical analysis and its applications. Inequalities containing fractional derivatives have applications in regard to fractional differential equations, especially in establishing the uniqueness of the solutions of initial value problems and their upper bounds. This kind of application motivated the researchers towards the theory of integral inequalities, with the aim of extending and generalizing classical inequalities using different fractional integral operators.
Theorem 2 ([18]). Let f : [a, b] → R be a convex function with f ∈ L 1 [a, b]. Then for σ > 0 Recall that the left-sided and the right-sided Riemann-Liouville fractional integrals of order σ > 0 are defined as in [19] for f ∈ L 1 [a, b] with Our aim is to prove Hermite-Hadamard's inequality in more general settings, and for this we need an extended generalized Mittag-Leffler function with its fractional integral operators and a class of (h, g; m)-convex functions.
The paper is structured as follows. In Section 2, we give present preliminary results and definitions that will be used in this paper. In Section 3, several Hermite-Hadamard-type inequalities for (h, g; m)-convex functions using fractional integral operators are presented. Furthermore, several properties and identities of these operators are given. As an application, in Section 4 we derive the upper bounds of fractional integral operators involving (h, g; m)-convex functions. In the last section, Section 5, we present the conclusions of this research.
More details on this generalized form of the Mittag-Leffler function and its fractional integral operators can be found in [1,2]. Here are some results we will use in this study: If we set a = 0 and x = 1 in (8), or b = 1 and x = 0 in (9), then we obtain the following corollary.
Setting α = 1 in Theorem 3, we obtain following identities for the constant function: ). Let the assumptions of Theorem 3 hold with α = 1. Then In this paper, we will use simplified notation to avoid a complicated manuscript form: . Of course, the conditions on all parameters ρ, σ, τ, ω, δ, c, q, r are essential and will be added to all theorems.

A Class of (h, g; m)-Convex Functions
Another direction for the generalization of the Hermite-Hadamard inequality is the use of different classes of convexity. For this we need a class of (h, g; m)-convex functions, the properties of which were recently presented in [14]: holds for all x, y ∈ I and all t ∈ (0, 1). If (12) holds in the reversed sense, then f is said to be an (h, g; m)-concave function.
This class unifies a certain range of convexity, enabling generalizations of known results. For different choices of functions h, g and parameter m, a class of (h, g; m)-convex functions is reduced to a class of P-functions [15], h-convex functions [17], m-convex functions [16], (h − m)-convex functions [11], (s, m)-Godunova-Levin functions of the second kind [10], exponentially s-convex functions in the second sense [9], etc. For example, if we set h(t) = t s , s ∈ (0, 1], g(x) = e −βx , β ∈ R, then we obtain a class defined in [13]: A function f : I ⊂ R → R is called exponentially (s, m)-convex in the second sense if the following inequality holds e βy m f (y) (13) for all x, y ∈ I and all t ∈ [0, 1], where β ∈ R, s, m ∈ (0, 1]. Next we need the Hermite-Hadamard inequality for (h, g; m)-convex functions:

Fractional Integral Inequalities of the Hermite-Hadamard Type for (h, g; m)-Convex Functions
The Hermite-Hadamard inequality for (h, g; m)-convex functions is obtained in [14], where some special results are pointed out and several known inequalities are improved upon. In [12], the article that followed, a few more inequalities of the Hermite-Hadamard type are presented. Here we will obtain their fractional generalizations, using (5)- (7), that is, the extended generalized Mittag-Leffler function E E E with fractional integral operators ε ε ε ω a + f and ε ε ε ω b − f in the real domain.
In this section, it is necessary to introduce the following conditions on the parameters and the interval [a, b]: We start with the left side, i.e., the first Hermite-Hadamard fractional integral inequality for (h, g; m)-convex functions involving the extended generalized Mittag-Leffler function. where Proof. Let f be an (h, g; m)-convex function on [0, ∞), m ∈ (0, 1]. Then for t = 1 2 we have Choosing y ≡ y m we obtain In the following step we will need to multiply both sides of the above inequality by t σ−1 E E E(ωt ρ ; p) and integrate on [0, 1] with respect to the variable t, which gives us Since m ∈ (0, 1], then a ≤ a/m, b ≤ b/m and [a, b] ⊂ [a, b m ]. Therefore, the condition f , g ∈ L 1 [a, b m ] is stated in this theorem. The above inequality can be written as Note that with Corollary 2 we can obtain the constant (ε ε ε ω a + 1)(b; p). This completes the proof.
Next we have the second Hermite-Hadamard fractional integral inequality.
Proof. Due to the (h, g; m)-convexity of f we have Multiplying both sides of above inequality by g(ta + (1 − t)b)t σ−1 E E E(ωt ρ ; p) and integrating on [0, 1] with respect to the variable t, we obtain With the substitution u = ta + (1 − t)b we obtain Again, due to the (h, g; m)-convexity of f we have Multiplying both sides of above inequality by g (1 − t) a m + t b m t σ−1 E E E(ωt ρ ; p) and integrating on [0, 1] with respect to the variable t, we obtain Inequality (17) now follows from (18) and (19).
In the following we derive fractional integral inequalities of Hermite-Hadamard type for different types of convexity, and state several corollaries, using special functions for h and/or g, and the parameter m. The first consequence of Theorems 5 and 6 obtained via the setting g ≡ 1 (i.e., g(x) = 1) is the Hermite-Hadamard fractional integral inequality for (h − m)-convex functions given in ( [20], Theorem 2.1): where ω and ω are defined by (16).
Proof. First we use substitutions t = b−x b−a and z = mx−a b−a in Theorem 6, after which we apply identities The result now follows from the above and Theorem 5.
By setting the function g ≡ 1 and the parameter m = 1, the previous result is reduced to the Hermite-Hadamard fractional integral inequality for h-convex functions: where ω is defined by (16).
In the following, we set the function h ≡ id, the identity function. With g ≡ 1 we obtain the Hermite-Hadamard fractional integral inequality for m-convex functions from ( where ω and ω are defined by (16).
The Hermite-Hadamard fractional integral inequality for convex functions is given in ( [21], Theorem 2.1). Here it is a merely a consequence for h ≡ id, g ≡ 1 and m = 1: Corollary 6. Let Assumption 1 hold. Let f be a nonnegative convex function on [0, ∞). If f ∈ L 1 [a, b], then the following inequalities hold where ω is defined by (16).

Proof.
Here we use We have presented several Hermite-Hadamard-type inequalities for the (h, g; m)convex function using fractional integral operators, where the kernel is an extended generalized Mittag-Leffler function. If we apply different parameter choices, as in Remark 2, then we obtain corresponding inequalities for different fractional operators.
Several Properties of Fractional Integral Operators ε ε ε ω a + f and ε ε ε ω b − f At the end of this section we give several results for fractional integral operators.
(ii) Furthermore, In particular, Note that (27) also follows directly from Corollary 2 if we set x = b in (10) and x = a in (11).
(ii) Equations (28) and (29) follow with the substitution z = a + b − t. Furthermore, (28) follows directly from Theorem 3 if we set x = b in (8) and x = a in (9). The final two equations are obtained for a = 0 and b = 1.

Remark 3.
To obtain the Hermite-Hadamard inequality for convex functions involving Riemann-Liouville fractional integrals, given in Theorem 2, first we need to set p = ω = 0 in (5) Note that a direct consequence of Theorem 3 is For the reader's convenience, we will directly prove this: Hence, from which follows .

Applications: Bounds of Fractional Integral Operators for (h, g; m)-Convex Functions
As an application, in this section we obtain the upper bounds of fractional integral operators for (h, g; m)-convex functions.
Theorem 8. Let Assumption 2 hold. If f , g ∈ L 1 [a, b] and h ∈ L 1 [0, 1], then for x ∈ [a, b] the following inequality holds where Proof. Using the proof follows analogously to that of Theorem 7.
From the two previous theorems we can directly obtain the following result.

Conclusions
This research was on Hermite-Hadamard-type inequalities existing in a more general setting. We used a fractional integral operator containing an extended generalized Mittag-Leffler function in the kernel, and obtained Hermite-Hadamard fractional integral inequalities for a class of (h, g; m)-convex functions. Furthermore, we presented the upper bounds of the fractional integral operators for (h, g; m)-convex functions. The obtained results generalize and extend the corresponding inequalities for different classes of convex functions.