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

In this paper, optimal convex combination bounds of centroidal and harmonic means for weighted geometric mean of logarithmic and identric means are proved. We find the greatest value lambda(alpha) and the least value Delta(alpha) for each alpha is an element of (0,1) such that the double inequality: lambda C(a, b) + (1 - lambda) H(a, b) < L-alpha (a, b)I1-alpha(a, b) < Delta C(a, b) + (1 - Delta)H(a, b) holds for all a, b > 0 with a not equal b. Here, C(a, b), H(a, b), L(a, b) and I(a, b) denote centroidal, harmonic, logarithmic and identric means of two positive numbers a and b, respectively.