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Some properties of the gamma and psi functions, with applications

Authors: S.-L. Qiu and M. Vuorinen
Journal: Math. Comp. 74 (2005), 723-742
MSC (2000): Primary 33B15; Secondary 26B15, 26D15, 51M25
Published electronically: May 18, 2004
MathSciNet review: 2114645
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Abstract: In this paper, some monotoneity and concavity properties of the gamma, beta and psi functions are obtained, from which several asymptotically sharp inequalities follow. Applying these properties, the authors improve some well-known results for the volume $\Omega_n$ of the unit ball $B^n\subset\mathbb{R}^n$, the surface area $\omega_{n-1}$ of the unit sphere $S^{n-1}$, and some related constants.

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Additional Information

S.-L. Qiu
Affiliation: President’s Office, Hangzhou Institute of Electronics Engineering (HIEE), Hangzhou 310037, Peoples Republic of China

M. Vuorinen
Affiliation: Department of Mathematics, University of Turku, Vesilinnankatu 5, FIN-20014, Turku, Finland

Keywords: Gamma function, beta function, psi function, monotoneity, concavity, inequalities
Received by editor(s): April 2, 2002
Received by editor(s) in revised form: September 27, 2003
Published electronically: May 18, 2004
Article copyright: © Copyright 2004 American Mathematical Society