On the Hermite-Hadamard type inequalities involving generalized integrals

In this paper, using a generalized integral operator, new Hermite-Hadamard type inequalities are obtained for diﬀerentiable modiﬁed ( h, m ) -convex functions of the second type.


Introduction
In recent years, integral inequalities became one of the most attractive areas in mathematics. Consequently, in this area, there has been a significant growth in the number of researchers and the findings gained in recent years. In this field of research, there is a classic inequality: the Hermite-Hadamard inequality (1) for convex functions, which is more than 130 years old and continues to attract mathematicians all over the world (for example, see [10,19,21]).
Let R, R + , and N be the sets of real numbers, positive real numbers, and positive integers, respectively. Also, we take R + 0 := R + ∪ {0} and N 0 := N ∪ {0}. If I ⊂ R is an interval and φ : I → R is a convex function, then for a, b ∈ I with a < b, the inequality holds. Convexity is a fundamental concept in geometry, but it is also utilized frequently in other areas of mathematics; for example, theory of optimization, theory of inequalities, functional analysis, mathematical programming, game theory, number theory, and variational calculus. The relationship between convexity and these branches becomes deeper and more beneficial day by day [14][15][16]41]. We recommend the paper [35] to readers who want a more comprehensive understanding of the many expansions and generalizations of the classical notion of convexity. The inequality (1) has become an object of research for many mathematicians, not only with the refinement of the classical concept of convexity, but also with the use of new integral operators, such as Riemann integral, fractional integrals of Riemann-Liouville type, and generalized integrals; for example, see [1, 3, 5, 8, 11-13, 20, 23, 25, 30, 33, 34, 36, 37, 44, 46] and the references cited therein.
The concept of m-convexity was introduced by Toader in [45]. The definition of an m-convex function is given as follows.
If (7) is reversed, then φ is said to be (h, m)-concave. In Definition 1.6, note that if h(t) = t then this definition coincides with the definition of an m-convex function; if in addition, we put m = 1 then we obtain the definition of a convex function. In [32], the authors presented the class of s-(a, m)-convex functions as follows ("redefined" in [47]).
holds, where (a, m) ∈ [0, 1] 2 and s ∈ (0, 1]. On the basis of the definitions listed before, we now present the classes of functions that are crucial for our main results (see [2]).

Remark 1.1.
Here, we list some special cases of Definition 1.8.
In the rest of this paper, we utilize the functions Γ (see [40,41,49,50]) and Γ k (see [10]) as defined below: It is noted here that Next, we provide some of the most well-known fractional operators (with the assumption that 0 ≤ ξ 1 < t < ξ 2 ≤ ∞) to make the main results of this paper easier to read. The well-known Riemann-Liouville fractional integrals are the first of these operators.
, then Riemann-Liouville fractional integrals of order α ∈ C, with (α) > 0, are defined by (right and left, respectively): The generalized integral operators that we use in this paper are defined in the next definition (see [18]).
In this paper, we present some variants of the inequality (1) for modified (h, m)-convex functions, within the framework of the generalized integral operators given in Definition 1.11.
Remark 2.1. In Theorem 2.1, if we consider the Riemann integral (or equivalently, if we take F ≡ 1 and ψ as a convex function (h(t) = t, s = 1, a = 1 and m = 1)), then from (19) we obtain the classical Hermite-Hadamard inequality (1). Also, this result is a variant of Theorem 9 of [38].
As we will see, the following result "complements" Theorem 2.1.

Remark 2.2. If
Of course, if we consider different kernels, then we get new variants of (20).
The next result is a more general variation of the previous two results, in which two modified (h, m)-convex functions of second type are involved. Theorem 2.3. Let ψ 1 be a modified (h 1 , m)-convex of the second type and ψ 2 be a modified (h 2 , m)-convex function of the second type such that ψ 1 ψ 2 ∈ L 1 [mξ 1 , mξ 2 ] and h 1 h 2 ∈ L 1 [ξ 1 , ξ 2 ]. The following inequality holds, where Proof. By using the definitions of the functions ψ 1 and ψ 2 , we have After multiplying and ordering, we get from (22) and (23) The desired inequality is obtained after integrating this last inequality, with respect to t between 0 and 1 and change of variables in the integrals of the left member. Remark 2.3. In Theorem 2.3, if we put F ≡ 1, s = 1, and if we consider only (22), then we obtain a complement to Theorem 2.2 of [31] for (h, m)-convex. If we consider the kernel F (t, α) = t 1−α , then we obtain new inequalities under Riemann-Liouville fractional integrals. If we use another kernel F , then we obtain inequalities not reported in the literature.
The next result gives a more general conclusion than Theorem 2.2.
Remark 2.4. In Theorem 2.3, if we take F ≡ 1, m = s = 1, and use only (25), then we get Theorem 2.3 of [31]. By using different kernels, we obtain new integral inequalities.

Conclusions
In this work, we have obtained several extensions and generalizations of the classical Hermite-Hadamard inequality, in the context of generalized integral operators. We have shown that several previously published results are particular cases of the ones that we have obtained. As a future work, it seems to be interesting to study other inequalities (for example, see [17]) by using generalized operators.