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

In this paper, we prove that the inequalities $\alpha [1/3 Q(a,b)+2/3 A(a,b)]+(1-\alpha)Q^{1/3}(a,b)A^{2/3}(a,b)<M(a,b) <\beta [1/3 Q(a,b)+2/3 A(a,b)]+(1-\beta)Q^{1/3}(a,b)A^{2/3}(a,b)$and$\lambda [1/6 C(a,b)+5/6 A(a,b)]+(1-\lambda)C^{1/6}(a,b)A^{5/6}(a,b)<M(a,b)<\mu [1/6 C(a,b)+5/6 A(a,b)]++(1-\mu)C^{1/6}(a,b)A^{5/6}(a,b)$holdforall$a,b>0$with$a\neq b$ifandonlyif$\alpha\leq (3-3\sqrt[6]{2}\log(1+\sqrt{2}))/[(2+\sqrt{2}-3\sqrt[6]{2})\log(1+\sqrt{2})]=0.777…$,$\beta\geq 4/5$,$\lambda\leq (6-6\sqrt[6]{2}\log(1+\sqrt{2}))/(7-6\sqrt[6]{2}\log(1+\sqrt{2}))=0.274…$, and $\mu \ge 8\mathrm{/}25$. Here, $M(a,b)$, $A(a,b)$, $C(a,b)$, and $Q(a,b)$ denote the Neuman-S'{a}ndor, arithmetic, contra-harmonic, and quadratic means of $a$ and $b$, respectively.