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

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 $a\mathrm{\ne }b$ if and only if $\alpha \ge 5\mathrm{/}9$ 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 $b$.