Optimal evaluation of a Toader-type mean by power mean

Abstract

In this paper, we present the best possible parameters $p, q\in\mathbb {R}$ such that the double inequality $M_{p}(a,b)< T[A(a,b), Q(a,b)]< M_{q}(a,b)$ holds for all $a, b>0$ with $a\neq b$ , and we get sharp bounds for the complete elliptic integral $\mathcal{E}(t)=\int _{0}^{\pi/2}(1-t^{2}\sin^{2}\theta)^{1/2}\,d\theta$ of the second kind on the interval $(0, \sqrt{2}/2)$ , where $T(a,b)=\frac{2}{\pi }\int _{0}^{\pi/2}\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}\,d\theta$ , $A(a,b)=(a+b)/2$ , $Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$ , $M_{r}(a,b)=[(a^{r}+b^{r})/2]^{1/r}$ ( $r\neq0$ ), and $M_{0}(a,b)=\sqrt {ab}$ are the Toader, arithmetic, quadratic, and rth power means of a and b, respectively.