On some inequality of Hermite-Hadamard type

TL;DR

This paper investigates inequalities related to the Hermite-Hadamard inequality in multivariate cases, revealing that certain classical inequalities do not extend straightforwardly and proposing optimal related inequalities.

Contribution

It introduces new inequalities for multivariate Hermite-Hadamard type and their Fejer counterparts, highlighting differences from the univariate case and proposing optimal bounds.

Findings

The classical inequality does not hold in multivariate cases.

New optimal inequalities for approximate integration are proposed.

Counterpart inequalities of Fejer type are established.

Abstract

It is well-known that the left term of the classical Hermite-Hadamard inequality is closer to the integral mean value than the right one. We show that in the multivariate case it is not true. Moreover, we introduce some related inequality comparing the methods of the approximate integration, which is optimal. We also present its counterpart of Fejer type.