Optimal bounds for the Neuman-Sandor means in terms of geometric and contra-harmonic means

TL;DR

This paper establishes precise bounds for the Neuman-Sandor mean using geometric and contra-harmonic means, providing exact conditions on the parameters for the inequalities to hold.

Contribution

It determines the optimal bounds for the Neuman-Sandor mean in terms of geometric and contra-harmonic means, with exact parameter conditions.

Findings

The inequality holds if and only if α ≥ 5/9 and β ≤ 0.4327.

Provides sharp bounds for the Neuman-Sandor mean.

Identifies the precise parameter range for the bounds.

Abstract

In this article, we prove that the double inequality $$\alpha G(a,b)+(1-\alpha)C(a,b)<M(a,b)<\beta G(a,b)+(1-\beta)C(a,b)$$ holds true for all $a,b>0$ with $a\neq b$ if and only if $\alpha\geq 5/9$ and $\beta\leq 1-1/[2\log(1+\sqrt{2})]=0.4327...$, where $G(a,b),C(a,b)$ and $M(a,b)$ are respectively the geometric, contra-harmonic and Neuman-S\'andor means of $a$ and $b$.