Strong h-Convexity and Separation Theorems

Teodoro Lara,Nelson Merentes,Kazimierz Nikodem

doi:10.1155/2016/7160348

Abstract

Jensen inequality for strongly h-convex functions and a characterization of pairs of functions that can be separated by a strongly h-convex function are presented. As a consequence, a stability result of the Hyers-Ulam type is obtained.

Highlights

Let X be a normed space, let D be a convex subset of X, and let c > 0
Combining the above two ideas we say that a function f : D → R is strongly h-convex with modulus c if f (tx + (1 − t) y) ≤ h (t) f (x) + h (1 − t) f (y)
The following result is a counterpart of the classical Jensen inequality for strongly h-convex functions

Summary

Introduction

Let X be a normed space, let D be a convex subset of X, and let c > 0. A function f : D → R is called strongly convex with modulus c (see, e.g., [1, 2]) if f (tx + (1 − t) y) ≤ tf (x) + (1 − t) f (y). Convex functions, introduced by Polyak [3], play an important role in optimization theory and mathematical economics. Combining the above two ideas we say that a function f : D → R is strongly h-convex with modulus c (cf [14]) if f (tx + (1 − t) y) ≤ h (t) f (x) + h (1 − t) f (y). Separation (or sandwich) theorems, that is, theorems providing conditions under which two given functions can be separated by a function from some special class, play an important role in many fields of mathematics and have various applications. In the literature one can find numerous results of this type (see, e.g., [12, 15,16,17,18,19,20,21,22,23,24])

Jensen-Type Inequality

Separation by Strongly h-Convex Functions