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

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]^λ[C^{1/6}(a,b)A^{5/6}(a,b)]^{1-λ}<M(a,b) &\qquad<[C(a,b)/6+5 A(a,b)/6]^μ[C^{1/6}(a,b)A^{5/6}(a,b)]^{1-μ} hold for all $a,b>0$ with $a\ne b$, where $M(a,b)$, $A(a,b)$ and $C(a,b)$ denote the Neuman-Sándor, arithmetic, and contra-harmonic means of $a$ and $b$, respectively.

9 pages