Refined inequalities on the weighted logarithmic mean

TL;DR

This paper presents new refined inequalities for convex functions, extending to weighted means like the logarithmic and identric mean, using a different approach from previous work.

Contribution

It introduces novel refined inequalities for convex functions and applies them to derive improved bounds for weighted logarithmic and identric means.

Findings

Refined inequalities for convex functions using Hermite-Hadamard inequality

New bounds for weighted logarithmic mean

Extensions to weighted identric mean

Abstract

Inspired by the recent work by R.Pal et al., we give further refined inequalities for a convex Riemann integrable function, applying the standard Hermite-Hadamard inequality. Our approach is different from their one in \cite{PSMA2016}. As corollaries, we give the refined inequalities on the weighted logarithmic mean and weighted identric mean. Some further extensions are also given.