Inequalities for Generalized Logarithmic Means
© Y.-M. Chu and W.-F. Xia. 2009
Received: 2 June 2009
Accepted: 10 December 2009
Published: 24 January 2010
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.
In , the following results are established: (1) implies that ; (2) implies that ; (3) implies that there exist such that ; (4) implies that there exist such that . Hence the question was answered: what are the least value and the greatest value such that the inequality holds for all ?
Stolarsky  proved that , with equality if and only if .
2. Main Results
This research is partly supported by N S Foundation of China under Grant 60850005 and N S Foundation of Zhejiang Province under Grants D7080080 and Y7080185.
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