The best bounds for Toader mean in terms of the centroidal and arithmetic means

TL;DR

This paper establishes optimal bounds for the Toader mean in terms of centroidal and arithmetic means, and applies these bounds to derive new estimates for the complete elliptic integral of the second kind.

Contribution

The paper determines the best constants for inequalities involving the Toader mean and classical means, providing new bounds and applications to elliptic integrals.

Findings

Derived the best constants for inequalities involving the Toader mean.

Established new bounds for the complete elliptic integral of the second kind.

Unified mean inequalities with optimal constants.

Abstract

In the paper, the authors discover the best constants $\alpha_{1}$, $\alpha_{2}$, $\beta_{1}$, and $\beta_{2}$ for the double inequalities $$ \alpha_{1}\bar{C}(a,b)+(1-\alpha_{1}) A(a,b)< T(a,b) <\beta_{1} \bar{C}(a,b)+(1-\beta_{1})A(a,b) $$ and $$ \frac{\alpha_{2}}{A(a,b)}+\frac{1-\alpha_{2}}{\bar{C}(a,b)}<\frac1{T(a,b)} <\frac{\beta_{2}}{A(a,b)}+\frac{1-\beta_{2}}{\bar{C}(a,b)} $$ to be valid for all $a,b>0$ with $a\ne b$, where $$ \bar{C}(a,b)=\frac{2(a^{2}+ab+b^{2})}{3(a+b)},\quad A(a,b)=\frac{a+b}2, $$ and $$ T(a,b)=\frac{2}{\pi}\int_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}\,\td\theta $$ are respectively the centroidal, arithmetic, and Toader means of two positive numbers $a$ and $b$. As an application of the above inequalities, the authors also find some new bounds for the complete elliptic integral of the second kind.