Sharp inequalities for the Neuman-Sandor mean in terms of arithmetic and contra-harmonic means

TL;DR

This paper establishes sharp bounds involving the Neuman-Sandor mean expressed through arithmetic and contra-harmonic means, determining optimal exponents for these inequalities.

Contribution

It derives the best possible constants and exponents for inequalities relating the Neuman-Sandor mean to arithmetic and contra-harmonic means.

Findings

Identified the greatest and least values of parameters for the inequalities.

Established sharp inequalities with optimal constants.

Provided a comprehensive analysis of mean relationships.

Abstract

In this paper, we find the greatest values $\alpha$ and $\lambda$, and the least values $\beta$ and $\mu$ such that the double inequalities $$C^{\alpha}(a,b)A^{1-\alpha}(a,b)<M(a,b)<C^{\beta}(a,b)A^{1-\beta}(a,b)$$ and &[C(a,b)/6+5 A(a,b)/6]^{\lambda}[C^{1/6}(a,b)A^{5/6}(a,b)]^{1-\lambda}<M(a,b) &\qquad<[C(a,b)/6+5 A(a,b)/6]^{\mu}[C^{1/6}(a,b)A^{5/6}(a,b)]^{1-\mu} hold for all $a,b>0$ with $a\neq b$, where $M(a,b)$, $A(a,b)$ and $C(a,b)$ denote the Neuman-S\'andor, arithmetic, and contra-harmonic means of $a$ and $b$, respectively.