Sharp Bounds for Neuman Means in Terms of Geometric, Arithemtic and Quadratic Means

Abstract

In this paper, we find the greatest values $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$, $\alpha_{4}$, $\alpha_{5}$, $\alpha_{6}$, $\alpha_{7}$, $\alpha_{8}$ and the least values $\beta_{1}$, $\beta_{2}$, $\beta_{3}$, $\beta_{4}$, $\beta_{5}$, $\beta_{6}$, $\beta_{7}$, $\beta_{8}$ such that the double inequalities $$A^{\alpha_{1}}(a,b)G^{1-\alpha_{1}}(a,b)<N_{GA}(a,b)<A^{\beta_{1}}(a,b)G^{1-\beta_{1}}(a,b),$$ $$\frac{\alpha_{2}}{G(a,b)}+\frac{1-\alpha_{2}}{A(a,b)}<\frac{1}{N_{GA}(a,b)}<\frac{\beta_{2}}{G(a,b)}+\frac{1-\beta_{2}}{A(a,b)},$$ $$A^{\alpha_{3}}(a,b)G^{1-\alpha_{3}}(a,b)<N_{AG}(a,b)<A^{\beta_{3}}(a,b)G^{1-\beta_{3}}(a,b),$$ $$\frac{\alpha_{4}}{G(a,b)}+\frac{1-\alpha_{4}}{A(a,b)}<\frac{1}{N_{AG}(a,b)}<\frac{\beta_{4}}{G(a,b)}+\frac{1-\beta_{4}}{A(a,b)},$$ $$Q^{\alpha_{5}}(a,b)A^{1-\alpha_{5}}(a,b)<N_{AQ}(a,b)<Q^{\beta_{5}}(a,b)A^{1-\beta_{5}}(a,b),$$ $$\frac{\alpha_{6}}{A(a,b)}+\frac{1-\alpha_{6}}{Q(a,b)}<\frac{1}{N_{AQ}(a,b)}<\frac{\beta_{6}}{A(a,b)}+\frac{1-\beta_{6}}{Q(a,b)},$$ $$Q^{\alpha_{7}}(a,b)A^{1-\alpha_{7}}(a,b)<N_{QA}(a,b)<Q^{\beta_{7}}(a,b)A^{1-\beta_{7}}(a,b),$$ $$\frac{\alpha_{8}}{A(a,b)}+\frac{1-\alpha_{8}}{Q(a,b)}<\frac{1}{N_{QA}(a,b)}<\frac{\beta_{8}}{A(a,b)}+\frac{1-\beta_{8}}{Q(a,b)}$$ hold for all $a, b>0$ with $a\neq b$, where $G$, $A$ and $Q$ are respectively the geometric, arithmetic and quadratic means, and $N_{GA}$, $N_{AG}$, $N_{AQ}$ and $N_{QA}$ are the Neuman means derived from the Schwab-Borchardt mean.