On an implicit convexity concept and some integral inequalities

Bessem Samet

doi:10.1186/s13660-016-1253-3

Abstract

We introduce a new concept of convexity that depends on a function $F:\mathbb{R}\times\mathbb{R}\times\mathbb {R}\times (0,1)\to\mathbb{R}$ satisfying certain axioms. The presented concept generalizes many kinds of convexity including e-convex functions, α-convex functions, and h-convex functions. Moreover, some integral inequalities are provided via our notion of convexity.

Highlights

Convexity is an important concept in many branches of mathematics, pure and applied
Many important integral inequalities are based on a convexity assumption of a certain function, such as Jensen’s inequality, the Hermite-Hadamard inequality, the Hardy-Littlewood-Pólya majoration inequality, Petrović’s inequality, Popoviciui’s convex function inequality, and many others
For more details as regards inequalities via convex functions, we refer the reader to the monograph [ ]

Summary

Introduction

Convexity is an important concept in many branches of mathematics, pure and applied. In particular, many important integral inequalities are based on a convexity assumption of a certain function, such as Jensen’s inequality, the Hermite-Hadamard inequality, the Hardy-Littlewood-Pólya majoration inequality, Petrović’s inequality, Popoviciui’s convex function inequality, and many others. We present a new concept of convexity that depends on a certain function satisfying some axioms. We introduce the new concept of convexity as follows. Taking ε = in the above example, we observe that any convex function f : [a, b] → R, (a, b) ∈ R , a < b, is an F-convex function with respect to the function F : R × R × R × ( , ) → R defined by. Let f : [a, b] → R, (a, b) ∈ R , a < b, be an F-convex function, for some F ∈ F.

The case of ε-convexity

The case of α-convexity

The case of h-convex functions