Inequalities between arithmetic-geometric, Gini, and Toader means (Q655612)

Summary: We find the greatest values \(p_1, p_2\) and least values \(q_1, q_2\) such that the double inequalities \(S_{p_1}(a, b) < M(a, b) < S_{q_1}(a, b)\) and \(S_{p_2}(a, b) < T(a, b) < S_{q_2}(a, b)\) hold for all \(a, b > 0\) with \(a \neq b\) 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.