Inequalities between Arithmetic-Geometric, Gini, and Toader Means

Abstract

We find the greatest values p1, p2 and least values q1, q2 such that the double inequalities Sp1(a,b)0 with a≠b and present some new bounds for the complete elliptic integrals. Here M(a,b), T(a,b), and Sp(a,b) are the arithmetic-geometric, Toader, and pth Gini means of two positive numbers a and b, respectively.