Another refinement of the right-hand side of the Hermite\u2013Hadamard inequality for simplices

Monika Nowicka,Alfred Witkowski

doi:10.1007/s00010-018-0595-y

Abstract

We establish a new refinement of the right-hand side of the Hermite–Hadamard inequality for convex functions of several variables defined on simplices.

Highlights

The classical Hermite–Hadamard inequality states that if f : I → R is a convex function for all a < b ∈ I the inequality f a+b 2
This powerful tool has found numerous applications and has been generalized in many directions. One of those directions is its multivariate version: Theorem 1. ([1]) Let f : U → R be a convex function defined on a convex set U ⊂ Rn and Δ ⊂ U be an n-dimensional simplex with vertices x0, x1, . . . , xn
X0 +···+xn n+1 is the barycenter of and the integration is with respect to the n-dimensional Lebesgue measure

Summary

Introduction

The classical Hermite–Hadamard inequality states that if f : I → R is a convex function for all a < b ∈ I the inequality f a+b 2. ([1]) Let f : U → R be a convex function defined on a convex set U ⊂ Rn and Δ ⊂ U be an n-dimensional simplex with vertices x0, x1, . X0 +···+xn n+1 is the barycenter of and the integration is with respect to the n-dimensional Lebesgue measure. The aim of this note is to prove a refinement of the right-hand side of (1) stated in Theorem 2.

Results

Conclusion