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
$$
\alpha_1\overline{C}(a,b)+(1-\alpha_1) A(a,b)< T(a,b) <\beta_1 \overline{C}(a,b)+(1-\beta_1)A(a,b)
$$
and
$$
\frac{\alpha_2}{A(a,b)}+\frac{1-\alpha_2}{\overline{C}(a,b)}<\frac1{T(a,b)} <\frac{\beta_2}{A(a,b)}+\frac{1-\beta_2}{\overline{C}(a,b)}
$$
to be valid for all $a,b>0$ with $a\ne b$, where
$$
\overline{C}(a,b)=\frac{2(a^2+ab+b^2)}{3(a+b)},\quad A(a,b)=\frac{a+b}2,
$$
and
$$
T(a,b)=\frac2{\pi}\int_0^{\pi/2}\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}\,\td\theta
$$
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.