The Hermite-Hadamard inequalities for $p$-convex functions

Abstract

In this paper, the Hermite-Hadamard inequality for p−convex function is provided. Some integral inequalities for them are also presented. Also, based on the integral and double integral of p−convex sets, the new functions are defined and under certain conditions, p−convexity of these functions are shown. Some inequalities for these functions are expressed.

Hermite-Hadamard type inequalities for $(p,h)$-convex functions on $\mathbb{R}^n$

“…In 1992, Dragomir [5] studied some properties of H and I for n = 1. In 2021, the authors [10] obtained the following conclusions for p-convex functions. (ii) For any t ∈ (0, 1], we have…”

“…The notation of the (p, h)-convex function generalizes some known classes of the usual pconvex function and the h-convex function, which are obtained by putting in (1.2) h(t) = t [24] and p = 1 [10], respectively. The h-convex function was introduced by Varosanec [24] and unifies the convex function, the s-convex function (in the second sense) [3], the Pfunction [21] and the Godunova-Levin function [12].…”

“…In 2021, Ekem, Kemali, Tnaztepe and Adilov [10] established the Hermite-Hadamard inequality for p-convex functions as follows.…”

“…[10]). Let f : R + → R + be an integrable p-convex function.For any [a, b] ⊂ R + , define the mappings H and I by (1.3) and (1.4) for n = 1, respectively.…”

Hermite-Hadamard Type Inequalities Related to s-Convex Functions in the Third Sense

“…In literature, the Hermite-Hadamard inequality for the generalizations of the convex functions is obtained by many researchers [4,[10][11][12]24]. As a continuation of these works, the extensions and refinements of the Hermite-Hadamard inequality for these functions satisfying certain condations have been the subject of many studies [5,13,15,17,[20][21][22]25].…”