Inequalities for means of two mean variables

TL;DR

This paper investigates the properties of various means such as Power, Identric, Lehmer, and Alzer means, focusing on their log-convexity and log-concavity, and explores their interrelations.

Contribution

It introduces new results on the log-convexity and log-concavity of multiple means and establishes their relationships, extending prior work by Anderson, Vamanamurthy, and Vuorinen.

Findings

Log-convexity and log-concavity of several means established

Relations among different means analyzed

Extensions of previous inequalities provided

Abstract

Motivated by the work of Anderson, Vamanamurthy and Vuorinen \cite{avv}, in this paper authors study the log-convexity and log-concavity of Power mean, Identric mean, weighted Power mean, Lehmer mean, Modified Alzer mean, and establish the relation of these means with each other.