Optimal power mean bounds for the weighted geometric mean of classical means. (English) Zbl 1187.26016
Summary: For , the power mean of order of two positive numbers and is defined by , for , and , for . In this paper, we answer the question: what are the greatest value and the least value such that the double inequality holds for all and with ? Here , , and denote the classical arithmetic, geometric, and harmonic means, respectively.
|26D15||Inequalities for sums, series and integrals of real functions|