Sharp inequalities for Toader mean in terms of other bivariate means

Abstract

In the paper, the author discover the best constants $\alpha_1$, $\alpha_2$, $\alpha_3$, $\beta_1$, $\beta_2$ and $\beta_3$ for the double inequalities
 \begin{equation*}
 \alpha_1 A\left(\frac{a-b}{a+b}\right)^{2n+2}