Sharp inequalities for Toader mean in terms of other bivariate means
Sharp inequalities for Toader mean in terms of other bivariate means
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.
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