Hermite–Hadamard Inequalities for Harmonic (s, m)-Convex Functions

Jian Zhong Xu, Umar Raza, Muhammad Waqas Javed, and Zaryab Hussain 4,5 Department of Electronics and Information Engineering, Bozhou University, Bozhou 236800, China Department of Mathematics, University of Jhang, Jhang 35200, Pakistan Department of Sciences and Humanities, National University of Computer and Emerging Sciences (FAST), Chiniot 35400, Pakistan Department of Mathematics, Punjab College of Commerce New Campus, Faisalabad 38000, Pakistan Department of Mathematics, Government College University Faislabad, Faisalabad 38000, Pakistan


Introduction and Preliminaries
Let C ⊂ R be an interval. en, C is said to be convex, if holds ∀ u, v ∈ C and t ∈ [0, 1]. Let C ⊂ R be an interval. en, a function f: C ⟶ R is said to be convex (concave), if holds ∀ u, v ∈ C and t ∈ [0, 1]. It can be easily seen in [1][2][3][4][5][6][7] that the convex (concave) functions have extensive applications in pure and applied mathematics, and in the literature [8][9][10][11][12][13][14][15], many eminent inequalities and other properties can be found in the framework of convexity. One of the renowned inequalities in the literature of Hermite-Hadamard Integral Inequality is given below: ese both inequalities hold in reverse if the function is concave. Now, the harmonic convex set is defined as follows.
Iscan [8] introduced the concept of harmonic convex function.
en, a function f: holds for all u, v ∈ C with (u, v) ≠ (0, 0) and t ∈ [0, 1]. In [8], Iscan by using the concept of harmonic convex function gave a new refinement of Hermite-Hadamard inequality as Definition 3. Let C ⊂ R be an interval and s, m ∈ (0, 1]. en, a function f: If s � m � 1, then harmonic (s, m)-convex function becomes the classical harmonic convex function. So harmonic convex function is a special case of harmonic (s, m)-convex function.
e main purpose of this article is to establish some conformable fractional estimates of Hermite-Hadamardtype inequalities via harmonic (s, m)-convex functions. Before going further towards our main results, let us have a brief review of the previously well known concepts and results. ese preliminaries will be highly helpful in acquiring the main results. e eminent gamma and beta functions are defined as e integral form of hypergeometric function is defined as For more details, see [11]. Recently, Abdeljawad [16] introduced the notation of right and left conformable fractional integrals for any positive order α as follows.

Main Results
In this section, we will present our main results.
Proof. By applying the definition of harmonic (s, m)-convex function for t � 1/2, we have Put x � uv/tu + (1 − t)v and y � uv/tv + (1 − t)u. en, We know that Now, consider a function f: R ⟶ R such that f(x) � 0. en, Also, So f is harmonic (s, m)-convex, and also, we have which implies that the inequality holds.
Here, fog is the composition function.
Proof. From inequality 1, we have 1 By using change of variable technique of integration, we have where g(x) � 1/x, and where g(x) � 1/x. Now, from (23), we have So the inequality holds.