Inequalities having Seven Means and Proportionality Relations

TL;DR

This paper explores inequalities involving seven classical means, establishing proportionality relations and inequalities among their differences, expanding the understanding of mean inequalities from a geometric perspective.

Contribution

It introduces new proportionality relations and inequalities among differences of seven means, extending prior geometric and inequality studies.

Findings

Established proportionality relations among seven means.

Derived inequalities among differences of these means.

Connected some means as special cases of Gini's mean.

Abstract

Eve (2003), studied seven means from geometrical point of view. These means are \textit{Harmonic, Geometric, Arithmetic, Heronian, Contra-harmonic, Root-mean square and Centroidal mean}. Some of these means are particular cases of Gini's (1938) mean of order r and s. In this paper we have established some proportionality relations having these means. Some inequalities among some of differences arising due to seven means inequalities are also established.