Some Integral Inequalities of Hermite-Hadamard Type for Convex Functions with Applications to Means

Bo-Yan Xi,Feng Qi

doi:10.1155/2012/980438

Abstract

The authors establish some new inequalities for differentiable convex functions, which are similar to the celebrated Hermite-Hadamard's integral inequality for convex functions, and apply these inequalities to construct inequalities for special means of two positive numbers.

Highlights

In 1, the following Hermite-Hadamard type inequalities for differentiable convex functions were proved.Theorem 1.1 see 1, Theorem 2.2
Adding these two equations leads to Lemma 2.1
This follows from a straightforward computation of definite integrals

Summary

Introduction

In 1 , the following Hermite-Hadamard type inequalities for differentiable convex functions were proved. Let f : I◦ ⊆ R → R be a differentiable mapping on I◦, a, b ∈ I◦ with a < b, and let p > 1. If the new mapping |f x |p/ p−1 is convex on a, b , f a f 2 b. Let f : a, b → R be an absolutely continuous mapping on a, b whose derivative belongs to Lp a, b. We will establish some new Hermite-Hadamard type integral inequalities for differentiable functions and apply them to derive some inequalities of special means

Lemmas

Some Integral Inequalities of Hermite-Hadamard Type

Applications to Means