- Research Article
- Open Access
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.
- Simple Computation
- Taylor Expansion
- Elementary Computation
- Monotonicity Result
- Generalize Logarithmic
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 .
for all , and . The upper bound in (1.6) is the best possible.
for all , where the function is the logarithmic derivative of the gamma function.
The purpose of this paper is to answer the following questions: what are the greatest values and , and the least value such that , , and for all ?
Equations (2.3) and (2.4) imply that for any there exists such that for .
Next we prove that is the optimal value for which the inequality holds.
Equations (2.7) and (2.8) imply that for any there exists such that for .
Equations (2.10) and (2.11) imply that for any there exists such that for .
Therefore, we cannot get inequality for any and all
It is easy to verify that for all
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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