A Sharp Double Inequality between Seiffert, Arithmetic, and Geometric Means

Abstract

For fixed $s\ge 1$ and any $t_1,t_2\in (\mathrm{0,1}/2)$ we prove that the double inequality holds for all $a,b>0$ with $a\ne b$ if and only if $t_1\le (1-\sqrt{1-(2/\pi {)}^{2/s}})/2$ and $t_2\ge (1-1/\sqrt{3s})/2$. Here, $P(a,b)$, $A(a,b)$ and $G(a,b)$ denote the Seiffert, arithmetic, and geometric means of two positive numbers $a$ and $b$, respectively.