Sandwich theorem for reciprocally strongly convex functions

Mireya Bracamonte,Jesús Medina,José Giménez

doi:10.15446/recolma.v52n2.77157

Abstract

We introduce the notion of reciprocally strongly convex functions and we present some examples and properties of them. We also prove that two real functions f and g, deﬁned on a real interval [a, b], satisfy for all x, y ∈ [a, b] and t ∈ [0, 1] iﬀ there exists a reciprocally strongly convex function h : [a, b] → R such that f (x) ≤ h(x) ≤ g(x) for all x ∈ [a, b]. Finally, we obtain an approximate convexity result for reciprocally strongly convex functions; namely we prove a stability result of Hyers-Ulam type for this class of functions.

Highlights

Due to its important role in mathematical economics, engineering, management science and optimization theory, convexity of functions and sets has been studied intensively; see [3, 7, 9, 11, 12, 13, 19, 22, 23] and the references therein
In [5] we proved the following sandwich theorem for harmonically convex functions: Theorem 2.4
In [4], we introduced the notion of harmonically strongly convex function, as follows: Definition 2.5

Summary

Introduction

Due to its important role in mathematical economics, engineering, management science and optimization theory, convexity of functions and sets has been studied intensively; see [3, 7, 9, 11, 12, 13, 19, 22, 23] and the references therein. In [5] we proved the following sandwich theorem for harmonically convex functions: Theorem 2.4. The following conditions are equivalent: (i) there exists a harmonically convex function h : (0, +∞) → R such that f (x) ≤ h (x) ≤ g (x), for all x ∈ (0, +∞); (ii) the following inequality holds xy f

Results

Conclusion