Functions Like Convex Functions

Zlatko Pavić

doi:10.1155/2015/919470

Abstract

The paper deals with convex sets, functions satisfying the global convexity property, and positive linear functionals. Jensen's type inequalities can be obtained by using convex combinations with the common center. Following the idea of the common center, the functional forms of Jensen's inequality are considered in this paper.

Highlights

Introduction is intended to be a brief overview of the concept of convexity and affinity
The paper deals with convex sets, functions satisfying the global convexity property, and positive linear functionals
A set S ⊆ X is convex if it contains all binomial convex combinations of its points

Summary

Introduction

Introduction is intended to be a brief overview of the concept of convexity and affinity. Let a, b ∈ X be points and let α, β ∈ R be coefficients. Their binomial combination αa + βb (1). A set S ⊆ X is convex if it contains all binomial convex combinations of its points. The convex hull convS of the set S is the smallest convex set containing S, and it consists of all binomial convex combinations of points of S. A function f : C → R is convex if the inequality f (αa + βb) ≤ αf (a) + βf (b). Holds for all binomial convex combinations αa + βb of pairs of points a, b ∈ C. Numerous papers have been written on Jensen’s inequality; different types and variants can be found in [2, 3]

Positive Linear Functionals and Convex Sets of Functions

Main Results

Applications to Functional Quasiarithmetic Means

Applications to Discrete and Integral Inequalities