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

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.