Refinements of the inequalities between Neuman-Sandor, arithmetic, contra-harmonic and quadratic means

Abstract

In this paper, we prove that the inequalities $\alpha [1/3Q(a,b)+2/3A(a,b)]+(1-\alpha )Q^{1/3}(a,b)A^{2/3}(a,b)<M(a,b)<\beta [1/3Q(a,b)+2/3A(a,b)]+(1-\beta )Q^{1/3}(a,b)A^{2/3}(a,b)$ and $\lambda [1/6C(a,b)+5/6A(a,b)]+(1-\lambda )C^{1/6}(a,b)A^{5/6}(a,b)<M(a,b)<\mu [1/6C(a,b)+5/6A(a,b)]++(1-\mu )C^{1/6}(a,b)A^{5/6}(a,b)$ hold for all $a,b>0$ with $a\ne b$ if and only if $\alpha \le (3-3\sqrt[6]{2}\log (1+\sqrt{2}))/[(2+\sqrt{2}-3\sqrt[6]{2})\log (1+\sqrt{2})]=\mathrm{0.777...}$, $\beta \ge 4/5$, $\lambda \le (6-6\sqrt[6]{2}\log (1+\sqrt{2}))/(7-6\sqrt[6]{2}\log (1+\sqrt{2}))=\mathrm{0.274...}$, and $\mu \ge 8/25$. Here, $M(a,b)$, $A(a,b)$, $C(a,b)$, and $Q(a,b)$ denote the Neuman-Sándor, arithmetic, contra-harmonic, and quadratic means of $a$ and $b$, respectively.

9 pages