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