Hermite-Hadamard Type Integral Inequalities for Functions Whose Second-Order Mixed Derivatives Are Coordinated(s,m)-P-Convex

Yu-Mei Bai,Ying Wu,Shan-He Wu

doi:10.1155/2018/1693075

Abstract

We establish some new Hermite-Hadamard type integral inequalities for functions whose second-order mixed derivatives are coordinated(s,m)-P-convex. An expression form of Hermite-Hadamard type integral inequalities via the beta function and the hypergeometric function is also presented. Our results provide a significant complement to the work of Wu et al. involving the Hermite-Hadamard type inequalities for coordinated(s,m)-P-convex functions in an earlier article.

Highlights

Let f : I → R be a convex mapping
Our results provide a significant complement to the work of Wu et al involving the Hermite-Hadamard type inequalities for coordinated (s, m)-P-convex functions in an earlier article
This celebrated inequality is known in the literature as the Hermite-Hadamard inequality

Summary

Introduction

Let f : I → R be a convex mapping. for any a, b ∈ I with a < b, we have the following double inequality: f(a + 2 b) ≤ b 1 − a b ∫ a f (x) dx f (a) + 2 f (b) . (1)This celebrated inequality is known in the literature as the Hermite-Hadamard inequality. We establish some new Hermite-Hadamard type integral inequalities for functions whose second-order mixed derivatives are coordinated (s, m)-P-convex. Our results provide a significant complement to the work of Wu et al involving the Hermite-Hadamard type inequalities for coordinated (s, m)-P-convex functions in an earlier article.

Results

Conclusion