and Applied Analysis 3 Alzer and Qiu 27 found the sharp bound of 1/2 L a, b I a, b in terms of the power mean as follows: Mc a, b < 1 2 L a, b I a, b 1.8 for all a, b > 0 with a/ b, with the best possible parameter c log 2/ 1 log 2 . The main purpose of this paper is to find the least value λ ∈ 0, 1 and the greatest value p p α such that αH a, b 1 − α L a, b > Mp a, b for α ∈ λ, 1 and all a, b ...