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

In this paper, we prove that the inequalities $α[1/3 Q(a,b)+2/3 A(a,b)]+(1-α)Q^{1/3}(a,b)A^{2/3}(a,b)<M(a,b) <β[1/3 Q(a,b)+2/3 A(a,b)]+(1-β)Q^{1/3}(a,b)A^{2/3}(a,b)$ and $λ[1/6 C(a,b)+5/6 A(a,b)]+(1-λ)C^{1/6}(a,b)A^{5/6}(a,b)<M(a,b)<μ[1/6 C(a,b)+5/6 A(a,b)]++(1-μ)C^{1/6}(a,b)A^{5/6}(a,b)$ hold for all $a,b>0$ with $a\neq b$ if and only if $α\leq (3-3\sqrt[6]{2}\log(1+\sqrt{2}))/[(2+\sqrt{2}-3\sqrt[6]{2})\log(1+\sqrt{2})]=0.777...$, $β\geq 4/5$, $λ\leq (6-6\sqrt[6]{2}\log(1+\sqrt{2}))/(7-6\sqrt[6]{2}\log(1+\sqrt{2}))=0.274...$, and $μ\geq 8/25$. 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.

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