OPTIMAL CONVEX COMBINATIONS BOUNDS OF CENTROIDAL AND HARMONIC MEANS FOR LOGARITHMIC AND IDENTRIC MEANS

We nd the greatest values 1 and 2, and the least values 1 and 2 such that the inequalities 1C(a;b)+(1 1)H(a;b) 0 with a6 b. Here, C(a;b), H(a;b), L(a;b), and I(a;b) are the centroidal, harmonic, logarithmic, and identric means of two positive numbers a and b, respectively.