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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On some properties of the Gamma function


Authors: Árpád Elbert and Andrea Laforgia
Journal: Proc. Amer. Math. Soc. 128 (2000), 2667-2673
MSC (2000): Primary 33B15; Secondary 26A48, 26D07
Published electronically: March 1, 2000
MathSciNet review: 1694859
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Abstract: Anderson and Qiu (1997) conjectured that the function $\frac{\log \Gamma (x+1)}{{x \log x}}$is concave for $x>1$. In this paper we prove this conjecture. We also study the monotonicity of some functions connected with the psi-function $\psi (x)$ and derive inequalities for $\psi (x)$ and $\psi '(x)$.


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Additional Information

Árpád Elbert
Affiliation: Mathematical Institute of the Hungarian Academy of Sciences, P.O. Box 127, Budapest H-1364, Hungary

Andrea Laforgia
Affiliation: Department of Mathematics, Largo S. Leonardo Murialdo, 1 00146 Roma, Italy
Email: laforgia@mat.uniroma3.it

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05520-9
PII: S 0002-9939(00)05520-9
Keywords: Gamma function, psi function, monotonicity, inequalities
Received by editor(s): October 23, 1998
Published electronically: March 1, 2000
Communicated by: Hal L. Smith
Article copyright: © Copyright 2000 American Mathematical Society