Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function

Abstract The primary objective of this research is to establish the generalized fractional integral inequalities of Hermite-Hadamard-type for MT-convex functions and to explore some new Hermite-Hadamard-type inequalities in a form of Riemann-Liouville fractional integrals as well as classical integrals. It is worth mentioning that our work generalizes and extends the results appeared in the literature.


Introduction
Integral inequality plays a critical role in both fields of pure and applied mathematics; see e.g. [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. It is clear that mathematical methods are useless without inequalities. For this reason, there is a present-day need for accurate inequalities in proving the existence and uniqueness of the mathematical methods. Besides, convexity plays a strong role in the field of inequalities due to the behaviour of its definition.
Let ⊆ and denote °by the interior of . A function → g : is said to be convex on , if the inequality ( + ( − ) ) ≤ ( ) + ( − ) ( ) g ζa ζ b ζg a ζ g b 1 1 (1) holds for all ∈ a b , and ∈ [ ] ζ 0, 1 . We say that g is concave, if −g is convex. Recently, a great number of equalities or inequalities for convex functions have been established by many authors. The representative results include Ostrowski-type inequality [15], Hardy-type inequality [16], Olsen-type inequality [17], Gagliardo-Nirenberg-type inequality [18], and the most well-known inequality of, namely, the Hermite-Hadamard-type inequality [19]. Here, we focus on it, which is formulated for a convex function ⊆ → g : by: with < a b and ∈ a b , , which can be a significant tool to obtain various priori estimates. Because of its importance, a number of scholars in the field of pure and applied mathematics have dedicated their efforts to extend, generalize, counterpart, and refine the Hermite-Hadamard inequality (2) for different classes of convex functions and mappings, see [20][21][22][23][24][25][26][27][28].
Definition 1.1. [29] A function ⊆ → g : is said to be MT-convex on , if it is nonnegative and satisfies the following inequality: , and ∈ ( ) ζ 0, 1 .
By virtue of the concept of MT-convexity, Park in [29] proved the following Hermite-Hadamard-type inequalities.
, with some > M 0. Then, for any ∈ ( ) h 0, 1 and > μ 0, the following inequality holds where β stands for the beta function of Euler type defined by , with > q 1, then we have where > p 1 is such that + = 1 p q , with some > M 0. Then, for any ∈ ( ) h 0,1 and > μ 0, the following inequality holds: For more details and interesting applications on Hermite-Hadamard-type inequalities for MT-convex functions, the reader is welcome to consult [30][31][32] and references therein. Furthermore, we recall the definition of generalized fractional integral operators by Sarikaya and Ertugral [33].
The left-and right-sided generalized fractional integral operators are defined as follows: respectively.
More recently, Qi et al. [40] established some inequalities of Hermite-Hadamard-type for ( ) α m , -convex functions by using generalized fractional integral operators (8) and (9). However, the main objective of this article is to explore several new and generalized fractional integral inequalities of Hermite-Hadamardtype for MT-convex functions involving generalized fractional integral operators (8) and (9).

Main results
At the beginning of the section, we first deliver an identity, which will play a significant role in the proof of our main results.
Proof. Integrating by parts and then using the equality = In particular, if ( ) = ρ ζ ζ , then we have the following corollary, which has been proved by Park [29].