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

TL;DR

This paper establishes sharp bounds for Neuman means in terms of geometric, arithmetic, and quadratic means, providing precise inequalities and bounds for these special means.

Contribution

It introduces the tightest possible bounds for Neuman means using classical means, extending the understanding of inequalities among these means.

Findings

Derived the greatest and least bounds for Neuman means in terms of classical means.

Established inequalities involving Neuman means and classical means with sharp bounds.

Provided explicit parameter ranges for inequalities to hold universally.

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.