Some properties of the gamma and psi functions, with applications

Qiu, S.-L.; Vuorinen, M.

doi:10.1090/S0025-5718-04-01675-8

Some properties of the gamma and psi functions, with applications
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by S.-L. Qiu and M. Vuorinen PDF

Math. Comp. 74 (2005), 723-742 Request permission

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.

References

Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1966. MR 0208797
Horst Alzer, Some gamma function inequalities, Math. Comp. 60 (1993), no. 201, 337–346. MR 1149288, DOI 10.1090/S0025-5718-1993-1149288-7
Horst Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997), no. 217, 373–389. MR 1388887, DOI 10.1090/S0025-5718-97-00807-7
H. Alzer, Inequalities for the gamma and polygamma functions, Abh. Math. Sem. Univ. Hamburg 68 (1998), 363–372. MR 1658358, DOI 10.1007/BF02942573
Horst Alzer, Inequalities for the volume of the unit ball in $\textbf {R}^n$, J. Math. Anal. Appl. 252 (2000), no. 1, 353–363. MR 1797860, DOI 10.1006/jmaa.2000.7065
H. Alzer: On Ramanujan’s double inequality for the gamma function (English. English summary), Bull. London Math. Soc. 35 (2003), no. 5, 601–607.
G. D. Anderson, R. W. Barnard, K. C. Richards, M. K. Vamanamurthy, and M. Vuorinen, Inequalities for zero-balanced hypergeometric functions, Trans. Amer. Math. Soc. 347 (1995), no. 5, 1713–1723. MR 1264800, DOI 10.1090/S0002-9947-1995-1264800-3
G. D. Anderson and S.-L. Qiu, A monotoneity property of the gamma function, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3355–3362. MR 1425110, DOI 10.1090/S0002-9939-97-04152-X
Glen D. Anderson, Mavina K. Vamanamurthy, and Matti K. Vuorinen, Conformal invariants, inequalities, and quasiconformal maps, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1997. With 1 IBM-PC floppy disk (3.5 inch; HD); A Wiley-Interscience Publication. MR 1462077
J. Heinonen, T. Kilpeläinen, and P. Koskela (eds.), Papers on analysis, Report. University of Jyväskylä Department of Mathematics and Statistics, vol. 83, University of Jyväskylä, Jyväskylä, 2001. A volume dedicated to Olli Martio on the occasion of his 60th birthday; Available electronically at http://www.math.jyu.fi/research/report83.html. MR 1886610, DOI 10.1007/bf03036358
George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
Johannes Böhm and Eike Hertel, Polyedergeometrie in $n$-dimensionalen Räumen konstanter Krümmung, Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften (LMW). Mathematische Reihe [Textbooks and Monographs in the Exact Sciences. Mathematical Series], vol. 70, Birkhäuser Verlag, Basel-Boston, Mass., 1981 (German). MR 626823
Christian Berg and Henrik L. Pedersen, A completely monotone function related to the gamma function, Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999), 2001, pp. 219–230. MR 1858281, DOI 10.1016/S0377-0427(00)00644-0
Karl-Heinz Borgwardt, The simplex method, Algorithms and Combinatorics: Study and Research Texts, vol. 1, Springer-Verlag, Berlin, 1987. A probabilistic analysis. MR 868467, DOI 10.1007/978-3-642-61578-8
Árpád Elbert and Andrea Laforgia, On some properties of the gamma function, Proc. Amer. Math. Soc. 128 (2000), no. 9, 2667–2673. MR 1694859, DOI 10.1090/S0002-9939-00-05520-9
Walter Gautschi, The incomplete gamma functions since Tricomi, Tricomi’s ideas and contemporary applied mathematics (Rome/Turin, 1997) Atti Convegni Lincei, vol. 147, Accad. Naz. Lincei, Rome, 1998, pp. 203–237. MR 1737497
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press, New York-London, 1965. Fourth edition prepared by Ju. V. Geronimus and M. Ju. Ceĭtlin; Translated from the Russian by Scripta Technica, Inc; Translation edited by Alan Jeffrey. MR 0197789
Ekatharine A. Karatsuba, On the computation of the Euler constant $\gamma$, Numer. Algorithms 24 (2000), no. 1-2, 83–97. Computational methods from rational approximation theory (Wilrijk, 1999). MR 1784993, DOI 10.1023/A:1019137125281
Ekatherina A. Karatsuba, On the asymptotic representation of the Euler gamma function by Ramanujan, J. Comput. Appl. Math. 135 (2001), no. 2, 225–240. MR 1850542, DOI 10.1016/S0377-0427(00)00586-0
D. Kershaw, Some extensions of W. Gautschi’s inequalities for the gamma function, Math. Comp. 41 (1983), no. 164, 607–611. MR 717706, DOI 10.1090/S0025-5718-1983-0717706-5
Daniel A. Klain and Gian-Carlo Rota, A continuous analogue of Sperner’s theorem, Comm. Pure Appl. Math. 50 (1997), no. 3, 205–223. MR 1431808, DOI 10.1002/(SICI)1097-0312(199703)50:3<205::AID-CPA1>3.0.CO;2-F
Andrea Laforgia, Further inequalities for the gamma function, Math. Comp. 42 (1984), no. 166, 597–600. MR 736455, DOI 10.1090/S0025-5718-1984-0736455-1
Ilija B. Lazarević and Alexandru Lupaş, Functional equations for Wallis and gamma functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 461-497 (1974), 245–251. MR 0361185
Milan Merkle, Logarithmic convexity and inequalities for the gamma function, J. Math. Anal. Appl. 203 (1996), no. 2, 369–380. MR 1410928, DOI 10.1006/jmaa.1996.0385
Milan Merkle, Convexity, Schur-convexity and bounds for the gamma function involving the digamma function, Rocky Mountain J. Math. 28 (1998), no. 3, 1053–1066. MR 1657048, DOI 10.1216/rmjm/1181071755
Milan Merkle, Conditions for convexity of a derivative and some applications to the gamma function, Aequationes Math. 55 (1998), no. 3, 273–280. MR 1615416, DOI 10.1007/s000100050036
D. S. Mitrinović, Analytic inequalities, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. MR 0274686, DOI 10.1007/978-3-642-99970-3
D. S. Mitrinović, J. Sándor, and B. Crstici, Handbook of number theory, Mathematics and its Applications, vol. 351, Kluwer Academic Publishers Group, Dordrecht, 1996. MR 1374329
J. Sandor, Sur la fonction gamma, Publ. Centre Rech. Math. Pures (I) (Neuchâtel), 21(1989), 4–7.
Nico M. Temme, Special functions, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996. An introduction to the classical functions of mathematical physics. MR 1376370, DOI 10.1002/9781118032572
S. R. Tims and J. A. Tyrrell, Approximate evaluation of Euler’s constant, Math. Gaz. 55 (1971), no. 391, 65–67. MR 491464, DOI 10.2307/3613323
Jussi Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR 0454009, DOI 10.1007/BFb0061216
Matti Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics, vol. 1319, Springer-Verlag, Berlin, 1988. MR 950174, DOI 10.1007/BFb0077904
E. T. Whittaker and G. N. Watson, A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions, Cambridge University Press, New York, 1962. Fourth edition. Reprinted. MR 0178117
R. M. Young, Euler’s constant, Math. Gaz., 75(1991), 187–190.