Sharp inequalities for Toader mean in terms of other bivariate means

Abstract

In the paper, the author discovers the best constants $\alpha_1$, $\alpha_2$, $\alpha_3$, $\beta_1$, $\beta_2$ and $\beta_3$ for the double inequalities \[ \alpha_1 A\left(\frac{a-b}{a+b}\right)^{2n+2} < T(a,b)-\frac{1}{4}C-\frac{3}{4}A-A\sum_{k=1}^{n-1}\frac{(\frac{1}{2},k)^2}{4((k+1)!)^2}\left(\frac{a-b}{a+b}\right)^{2k+2}<\beta_1 A\left(\frac{a-b}{a+b}\right)^{2n+2} \] \[ \alpha_2 A\left(\frac{a-b}{a+b}\right)^{2n+2} < T(a,b)-\frac{3}{4}\overline{C}-\frac{1}{4}A-A\sum_{k=1}^{n-1}\frac{(\frac{1}{2},k)^2}{4((k+1)!)^2}\left(\frac{a-b}{a+b}\right)^{2k+2}<\beta_2 A\left(\frac{a-b}{a+b}\right)^{2n+2} \] and \[ \alpha_3 A\left(\frac{a-b}{a+b}\right)^{2n+2} < \frac{4}{5}T(a,b)+\frac{1}{5}H-A-A\sum_{k=1}^{n-1}\frac{(\frac{1}{2},k)^2}{5((k+1)!)^2}\left(\frac{a-b}{a+b}\right)^{2k+2}<\beta_3 A\left(\frac{a-b}{a+b}\right)^{2n+2} \] to be valid for all $a,b>0$ with $a\ne b$ and $n=1,2,\cdots$, where \[ C\equiv C(a,b)=\frac{a^2+b^2}{a+b},\,\overline{C}\equiv\overline{C}(a,b)=\frac{2(a^2+ab+b^2)}{3(a+b)},\, A\equiv A(a,b)=\frac{a+b}{2}, \] \[ H\equiv H(a,b) =\frac{2ab}{a+b},\quad T(a,b)=\frac2{\pi}\int_0^{\pi/2}\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}\,{\rm d}\theta \] are respectively the contraharmonic, centroidal, arithmetic, harmonic and Toader means of two positive numbers $a$ and $b$, $ (a,n)=a(a+1)(a+2)(a+3)\cdots (a+n-1)$ denotes the shifted factorial function. As an application of the above inequalities, the author also find a new bounds for the complete elliptic integral of the second kind.