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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
DOI: https://doi.org/10.1090/S0025-5718-04-01675-8
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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  • [AS] M. ABRAMOWITZ AND I. A. STEGUN, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1965. MR 34:8606
  • [A1] H. ALZER, Some gamma function inequalities, Math. Comp., 60(1993), 337-346. MR 93f:33001
  • [A2] H. ALZER, On some inequalities for the gamma and psi functions, Math. Comp., 66(1997), 373-389. MR 97e:33004
  • [A3] H. ALZER, Inequalities for the gamma and polygamma functions, Abh. Math. Sem. Univ. Hamburg, 68(1998), 363-372. MR 99k:33002
  • [A4] H. ALZER: Inequalities for the volume of the unit ball in $\mathbb{R}^n$, J. Math. Anal. Appl. 252 (2000), 353-363. MR 2001m:26036
  • [A5] H. ALZER: On Ramanujan's double inequality for the gamma function (English. English summary), Bull. London Math. Soc. 35 (2003), no. 5, 601-607.
  • [ABRVV] 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), 1713-1723. MR 95m:33002
  • [AQ] G. D. ANDERSON AND S.-L. QIU, A monotoneity property of the gamma function, Proc. Amer. Math. Soc., (125)1995, 3355-3362. MR 98h:33001
  • [AVV1] G. D. ANDERSON, M. K. VAMANAMURTHY AND M. VUORINEN, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, 1997. MR 98h:30033
  • [AVV2] G. D. ANDERSON, M. K. VAMANAMURTHY, AND M. VUORINEN, Topics in special functions, Papers on Analysis: A volume dedicated to Olli Martio on the occasion of his 60th birthday, ed. by J. Heinonen, T. Kilpeläinen, and P. Koskela, Report 83, Univ. Jyväskylä (2001), 5-26, ISBN 951-39-1120-9. (http://www.math.jyu.fi/research/report83.html) MR 2002j:00014
  • [AAR] G. ANDREWS, R. ASKEY, R. ROY, Special Functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge Univ. Press, 1999. MR 2000g:33001
  • [BH] J. B¨OHM AND E. HERTEL, Polyedergeometrie $n$-dimensionalen Räumen Konstanter Krümmung, Birkhäuser, Basel-Boston-Stuttgart, 1981. MR 82k:52001a
  • [BP] C. BERG AND H. PEDERSEN, A completely monotone function related to the gamma function, J. Comp. Appl. Math. 133 (2001), 219-230. MR 2003k:33001
  • [B] K. H. BORGWARDT, The Simplex Method, a Probabilistic Analysis, Springer-Verlag, Berlin, 1987. MR 88k:90110
  • [EL] Á. ELBERT AND A. LAFORGIA, On some properties of the gamma function, Proc. Amer. Math. Soc. 128 (2000), 2667-2673.MR 2000m:33002
  • [G] W. GAUTSCHI, The incomplete gamma functions since Tricomi. Tricomi's ideas and contemporary applied mathematics (Rome/Turin, 1997), 203-237, Atti Convegni Lincei, 147, Accad. Naz. Lincei, Rome, 1998. MR 2001g:33003
  • [GR] I. S. GRADSHTEYN AND I. M. RYZHIK, Tables of Integrals, Series, and Products, 4th ed., prepared by Yu. V. Geronimus and M. Yu. Tseytlin, Academic Press, New York-London, 1965. MR 33:5952
  • [K1] E. A. KARATSUBA, On the computation of the Euler constant $\gamma$, Numer. Algorithms 24 (2000), 83-87. MR 2002f:33004
  • [K2] E. A. KARATSUBA: On the asymptotic representation of the Euler gamma function by Ramanujan, J. Comp. Appl. Math. 135.2 (2001), 225-240.MR 2002i:33004
  • [Ke] D. KERSHAW, Some extensions of W. Gautschi's inequalities for the gamma function. Math. Comp. 41 (1983), no. 164, 607-611.MR 84m:33003
  • [KR] D. A. KLAIN AND G.-C. ROTA, A continuous analogue of Sperner's Theorem, in Communications on Pure and Applied Mathematics, Vol. L, John Wiley & Sons, 1997, pp. 205-223.MR 98c:05155
  • [L] A. LAFORGIA, Further inequalities for the gamma function, Math. Comp. 42 (1984), no. 166, 597-600. MR 85i:33001
  • [LL] I. B. LAZAREVIC AND A. LUPAS, Functional equations for Wallis and gamma functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Electr. Telec. Autom, 461-497(1974), 245-251. MR 50:13631
  • [Me1] M. MERKLE, Logarithmic convexity and inequalities for the gamma function, J. Math. Anal. Appl., 203(1996), 369-380.MR 98c:33001
  • [Me2] M. 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 99m:33001
  • [Me3] M. MERKLE, Conditions for convexity of a derivative and some applications to the gamma function, Aequationes Math. 55 (1998), no. 3, 273-280. MR 99d:33002
  • [Mi] D. S. MITRINOVIC, Analytic Inequalities, Grundlehren Math. Wiss., Band 165, Springer-Verlag, Berlin, 1970. MR 43:448
  • [MSC] D. S. MITRINOVIC, J. SANDOR AND B. CRSTICI, Handbook of Number Theory, Kluwer Acad. Publ., 1995. MR 97f:11001
  • [S] J. SANDOR, Sur la fonction gamma, Publ. Centre Rech. Math. Pures (I) (Neuchâtel), 21(1989), 4-7.
  • [T] N. M. TEMME, Special functions. An introduction to the classical functions of mathematical physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996. MR 97e:33002
  • [TT] S. R. TIMS AND J. A. TYRRELL, Approximate evaluation of Euler's constant, Math. Gaz., 55(1971), 65-67. MR 58:10710
  • [V] J. V¨AISÄLÄ, Lectures on $n$-Dimensional Quasiconformal Mappings, Lecture Notes in Math., Vol.229, Springer-Verlag, Berlin, 1971. MR 56:12260
  • [Vu] M. VUORINEN, Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math., Vol. 1319, Springer-Verlag, Berlin, 1988. MR 89k:30021
  • [WW] E. T. WHITTAKER AND G. N. WATSON, A Course of Modern Analysis, 4th ed., Cambridge Univ. Press, London, 1958. MR 31:2375
  • [Y] R. M. YOUNG, Euler's constant, Math. Gaz., 75(1991), 187-190.

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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
Email: sl_qiu@hziee.edu.cn

M. Vuorinen
Affiliation: Department of Mathematics, University of Turku, Vesilinnankatu 5, FIN-20014, Turku, Finland
Email: vuorinen@csc.fi

DOI: https://doi.org/10.1090/S0025-5718-04-01675-8
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

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