Sequences of Inequalities among Differences of Gini Means and Divergence Measures

TL;DR

This paper investigates inequalities among Gini means and divergence measures, analyzing their convexity and establishing new inequality sequences, with connections to probability distributions and divergence measures.

Contribution

It introduces new sequences of inequalities among Gini means, studies their convexity properties, and links these differences to well-known divergence measures in probability theory.

Findings

Identified convex and non-convex differences among Gini means.

Established new inequality sequences involving Gini means.

Connected differences to classical divergence measures.

Abstract

In 1938, Gini studied a mean having two parameters. Later, many authors studied properties of this mean. In particular, it contains the famous means as harmonic, geometric, arithmetic, etc. Here we considered a sequence of inequalities arising due to particular values of each parameter of Gini's mean. This sequence generates many nonnegative differences. Not all of them are convex. We have studied here convexity of these differences and again established new sequences of inequalities of these differences. Considering in terms of probability distributions these differences, we have made connections with some of well known divergence measures.