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

TL;DR

This paper refines inequalities involving Neuman-Sandor, arithmetic, contra-harmonic, and quadratic means, establishing precise bounds and conditions for their relationships.

Contribution

It provides new sharp bounds and necessary and sufficient conditions for inequalities between these classical means.

Findings

Derived explicit bounds for mean inequalities.

Established conditions for the inequalities to hold.

Enhanced understanding of relationships between classical means.

Abstract

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.