An extension of Jensen's discrete inequality to half convex functions

Vasile Cirtoaje,Alina Baiesu

doi:10.1186/1029-242x-2011-101

Abstract

We extend the right and left convex function theorems to weighted Jensen's type inequalities, and then combine the new theorems in a single one applicable to a half convex function f(u), defined on a real interval and convex for u ≤ s or u ≥ s, where . The obtained results are applied for proving some open relevant inequalities.

Highlights

The right convex function theorem (RCF-Theorem) has the following statement.RCF-Theorem
Let f(u) be a function defined on a real interval Á and convex for u ≥ s ∈ Á
Let f(u) be a function defined on a real interval Á and convex for u ≤ s ∈ Á

Summary

Introduction

The right convex function theorem (RCF-Theorem) has the following statement (see [1,2,3]). Applying RCF-, LCF-, and HCF-Theorems to the function f(u) = g(eu), and replacing s by ln r, x by ln x, y by ln y, and each xi by ln ai for i = 1, 2, ..., n, we get the following corollaries, respectively. On the other hand, applying WRCF-, WLCF-, and WHCF-Theorems to the function f (u) = g(eu) and replacing s by 1nr, x by 1nx, y by 1ny, and each xi by In ai for i = 1, 2, ... WRCF-Corollary Let g be a function defined on a positive interval Á such that f(u) = g(eu) is convex for ln u ≥ r ∈ Á, and let p1, p2, ..., pn be positive real numbers such that p = min{p1, p2, . The required inequalities in WRCF-, WLCF-, and WHCF-Theorems turn into equalities for x1 = x2 = ...

From λi

For p