Copyright © 2009 Yu-Ming Chu and Wei-Feng Xia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
For , the generalized logarithmic mean of two positive numbers and is defined as , for ,
, for , , , , for , , and
, for , . In this paper, we prove that , and for all , and the constants , and cannot be improved for the corresponding inequalities. Here , and denote the arithmetic, geometric, and harmonic means of and , respectively.