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

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