Inequalities between Arithmetic-Geometric, Gini, and Toader Means

Abstract

We find the greatest values $p_1$, $p_2$ and least values $q_1$, $q_2$ such that the double inequalities and hold for all $a,b>0$ with $a\ne b$ and present some new bounds for the complete elliptic integrals. Here $M(a,b)$, $T(a,b)$, and $S_p(a,b)$ are the arithmetic-geometric, Toader, and $p$th Gini means of two positive numbers $a$ and $b$, respectively.