Hermite-Hadamard-like type inequalities for twice differentiable MT-convex functions

Abstract

In this paper, we establish some new Hermite-Hadamard-like type inequalities for continuously twice differentiable (α, m)-geometric-arithmetically convex functions. Mathematics Subject Classification: 26A51, 26D15Keywords: Hermite-Hadamard-type inequality, Hölder's inequality, Pólya inequality, (α, m)-GA-convexityare respectively called the weighted geometric mean of two positive numbers x and y and the weighted arithmetic mean of f (x) and f (y). Recently, in [5, 8], Ji et al. introduced the concepts of (α, m)-geometricarithmetically convex function as follows: Definition 2. The function f : [0, b] → R is said to be (α, m)-geometricarithmetically convex, or simply speaking, (α, m)-GA-convex, where (α, m) ∈ [0, 1] 2 , if the inequality f (x t y m(1−t) ) ≤ t α f (x) + m(1 − t α )f (y) holds, for all x, y ∈ [0, b] and t ∈ [0, 1]. Then, also Ji et al.[5] obtained the Hermite-Hadamard type inequalities for (α, m)-GA-convex function as follows: Theorem 1.1. Let f : R 0 = [0, ∞) → R be a differentiable function and f ∈ L([a, b]) for 0 < a < b < ∞. If |f | q is an (α, m)-geometric-arithmetically convex function on [0, max{a

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