Hermite-Hadamard type inequalities for n-times differentiable and preinvex functions

Shu-Hong Wang,Feng Qi

doi:10.1186/1029-242x-2014-49

Abstract

In the paper, by creating an integral identity involving an n-times differentiable function, the authors establish some new Hermite-Hadamard type inequalities for preinvex functions and generalize some known results.

Highlights

Throughout this paper, let R = (–∞, ∞) and N denote the set of all positive integers.Let us recall some definitions of various convex functions.Definition A function f : I ⊆ R → R is said to be convex if f λx + ( – λ)y ≤ λf (x) + ( – λ)f (y) ( )holds for all x, y ∈ I and λ ∈ [, ]
Definition [ ] Let S ⊆ Rn be an invex set with respect to η : S × S → Rn
A function f : S → R is said to be preinvex with respect to η, if f (y + tη(x, y)) ≤ tf (x) + ( – t)f (y) for every x, y ∈ S and t ∈ [, ]

Summary

Introduction

Definition [ ] Let S ⊆ Rn be an invex set with respect to η : S × S → Rn. A function f : S → R is said to be preinvex with respect to η, if f (y + tη(x, y)) ≤ tf (x) + ( – t)f (y) for every x, y ∈ S and t ∈ [ , ].

Results

Conclusion