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 $a\ne b$, 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.

11 pages