The Best Bounds for Toader Mean in Terms of the Centroidal and Arithmetic Means

Yun Hua, Feng Qi

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 \[ \frac{\alpha_2}{A(a,b)}+\frac{1-\alpha_2}{\bar C(a,b)}<\frac1{T(a,b)}<\frac{\beta_2}{A(a,b)}+\frac{1-\beta_2}{\bar C(a,b)} \] to be valid for all $a,b>0$ with $a\neq b$, where \[ \bar C(a,b)=\frac{2(a^2+ab+b^2)}{3(a+b)}, \ A(a,b)=\frac{a+b}2,\text { and } T(a,b)=\frac2\pi\int^{\pi/2}_0\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}\text d\theta \] 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.