Optimal Inequalities between Harmonic, Geometric, Logarithmic, and Arithmetic-Geometric Means

We find the least values p, q, and s in (0, 1/2) such that the inequalities H(pa+(1 − p)b, pb+(1 − p)a)>AG(a,b), G(qa+(1−q)b, qb+(1−q)a)>AG(a,b), and L(sa+(1−s)b,sb+(1−s)a)> AG(a,b) hold for all a,b>0 with a≠b, respectively. Here AG(a,b), H(a,b), G(a,b), and L(a,b) denote the arithmetic-geometric, harmonic, geometric, and logarithmic means of two positive numbers a and b, respectively

Inequalities between Arithmetic-Geometric, Gini, and Toader Means

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