MMN-1775
Bounds for the arithmetic mean in terms of the Toader mean and other bivariate means
Yun Hua;Abstract
We find the greatest value
$\alpha_{1}$ and $\alpha_{2}$, and the least values $\beta_{1}$ and
$\beta_{2}$, such that the double inequalities $\alpha_{1}
T(a,b)+(1-\alpha_{1}) H(a,b)0$ with
$a\neq b$. Here, $\overline{H}(a,b)=\sqrt{2}ab/\sqrt{a^2+b^2}$, $H(a,b)=2ab/(a+b)$, $A(a,b)=(a+b)/2$, and
$T(a,b)=\frac{2}{\pi}\int\limits_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}d\theta$
denote the harmonic root-square, harmonic, arithmetic and Toader means of $a$ and $b$, respectively.
Vol. 18 (2017), No. 1, pp. 203-210