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 $$\frac{{\alpha }_2}{A(a,b)}+\frac{1-{\alpha }_2}{\bar{C}(a,b)}<\frac{1}{T(a,b)}<\frac{{\beta }_2}{A(a,b)}+\frac{1-{\beta }_2}{\bar{C}(a,b)}$$ to be valid for all $a,b>0$ with $a\ne b$, where $$\bar{C}(a,b)=\frac{2(a^2+ab+b^2)}{3(a+b)},A(a,b)=\frac{a+b}{2},$$ and $$ T(a,b)=\frac{2}π\int_{0}^{π/{2}}\sqrt{a^2{\cos^2θ}+b^2{\sin^2θ}}\,\tdθ$$ 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.

7 pages

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