A unified proof of inequalities and some new inequalities involving Neuman-S\'andor mean

TL;DR

This paper provides a unified proof for inequalities involving the Neuman-Sándor mean and introduces new sharp inequalities relating it to other classical means of positive numbers.

Contribution

It offers a unified proof approach for existing inequalities and presents new sharp inequalities involving the Neuman-Sándor mean and other means.

Findings

Unified proof of inequalities for Neuman-Sándor mean

New sharp inequalities involving multiple means

Bounding Neuman-Sándor mean with classical means

Abstract

In the paper, by finding linear relations of differences between some means, the authors supply a unified proof of some double inequalities for bounding Neuman-S\'andor means in terms of the arithmetic, harmonic, and contra-harmonic means and discover some new sharp inequalities involving Neuman-S\'andor, contra-harmonic, root-square, and other means of two positive real numbers.