Inequalities for the gamma function
Author:
Horst Alzer
Journal:
Proc. Amer. Math. Soc. 128 (2000), 141147
MSC (1991):
Primary 33B15; Secondary 26D07
Published electronically:
June 30, 1999
MathSciNet review:
1622757
Fulltext PDF Free Access
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Abstract: We prove the following two theorems: (i) Let be the th power mean of and . The inequality holds for all if and only if , where denotes Euler's constant. This refines results established by W. Gautschi (1974) and the author (1997). (ii) The inequalities are valid for all if and only if and , while holds for all if and only if and . These bounds for improve those given by G. D. Anderson an S.L. Qiu (1997).
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 H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997), 373389. MR 97e:33004
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 W. Gautschi, A harmonic mean inequality for the gamma function SIAM J. Math. Anal. 5 (1974), 278281. MR 50:2570
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 W. Gautschi, Some mean value inequalities for the gamma function, SIAM J. Math. Anal. 5 (1974), 282292. MR 50:2571
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Additional Information
Horst Alzer
Affiliation:
Morsbacher Str. 10, 51545 Waldbröl, Germany
DOI:
http://dx.doi.org/10.1090/S000299399904993X
PII:
S 00029939(99)04993X
Keywords:
Gamma function,
psi function,
power mean,
inequalities
Received by editor(s):
March 10, 1998
Published electronically:
June 30, 1999
Communicated by:
Hal L. Smith
Article copyright:
© Copyright 1999
American Mathematical Society
