OPTIMAL GENERALIZED LOGARITHMIC MEAN BOUNDS FOR THE GEOMETRIC COMBINATION OF ARITHMETIC AND HARMONIC MEANS

In this paper, we answer the question: for alpha is an element of (0,1), what are the greatest value p = p (alpha) and least value q = q (alpha), such that the double inequality L-p(a, b) <= A(alpha)(a, b)H1-alpha(a, b) <= L-q (a, b) holds for all a, b > 0? where L-p (a, b), A(a, b), and H(a, b) are the p-th generalized logarithmic, arithmetic, and harmonic means of a and b, respectively.