  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 $\lambda [1/6 C(a,b)+5/6
A(a,b)]+(1-\lambda)C^{1/6}(a,b)A^{5/6}(a,b)<M(a,b)<\mu [1/6 C(a,b)+5/6
A(a,b)]++(1-\mu)C^{1/6}(a,b)A^{5/6}(a,b)$ 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 the
Neuman-S\'{a}ndor, arithmetic, contra-harmonic, and quadratic means of $a$ and
$b$, respectively.
