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

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 $a\ne b$ if and only if $\alpha \ge 5/9$ and $\beta \le 1-1/[2\log (1+\sqrt{2})]=\mathrm{0.4327...}$, where $G(a,b),C(a,b)$ and $M(a,b)$ are respectively the geometric, contra-harmonic and Neuman-Sándor means of $a$ and $b$.

6 pages