A one-parameter family of Pick functions defined by the Gamma function and related to the volume of the unit ball in $n$-space
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- by Christian Berg and Henrik L. Pedersen PDF
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Abstract:
We show that \[ F_a(x)=\frac {\ln \Gamma (x+1)}{x\ln (ax)} \] can be considered as a Pick function when $a\ge 1$, i.e. extends to a holomorphic function mapping the upper half-plane into itself. We also consider the function \[ f(x)=\left (\frac {\pi ^{x/2}}{\Gamma (1+x/2)}\right )^{1/(x\ln x)} \] and show that $\ln f(x+1)$ is a Stieltjes function and that $f(x+1)$ is completely monotonic on $]0,\infty [$. In particular, $f(n)=\Omega _n^{1/(n\ln n)},n\ge 2$, is a Hausdorff moment sequence. Here $\Omega _n$ is the volume of the unit ball in Euclidean $n$-space.References
- N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965. Translated by N. Kemmer. MR 0184042
- Horst Alzer, Inequalities for the volume of the unit ball in $\Bbb R^n$. II, Mediterr. J. Math. 5 (2008), no. 4, 395–413. MR 2465568, DOI 10.1007/s00009-008-0158-x
- G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Special functions of quasiconformal theory, Exposition. Math. 7 (1989), no. 2, 97–136. MR 1001253
- G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Topics in special functions, Papers on analysis, Rep. Univ. Jyväskylä Dep. Math. Stat., vol. 83, Univ. Jyväskylä, Jyväskylä, 2001, pp. 5–26. MR 1886609
- 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
- Emil Artin, The gamma function, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York-Toronto-London, 1964. Translated by Michael Butler. MR 0165148
- Christian Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), no. 4, 433–439. MR 2112748, DOI 10.1007/s00009-004-0022-6
- Christian Berg, Jens Peter Reus Christensen, and Paul Ressel, Harmonic analysis on semigroups, Graduate Texts in Mathematics, vol. 100, Springer-Verlag, New York, 1984. Theory of positive definite and related functions. MR 747302, DOI 10.1007/978-1-4612-1128-0
- Christian Berg and Gunnar Forst, Potential theory on locally compact abelian groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 87, Springer-Verlag, New York-Heidelberg, 1975. MR 0481057
- 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
- Christian Berg and Henrik L. Pedersen, Pick functions related to the gamma function, Rocky Mountain J. Math. 32 (2002), no. 2, 507–525. Conference on Special Functions (Tempe, AZ, 2000). MR 1934903, DOI 10.1216/rmjm/1030539684
- William F. Donoghue Jr., Monotone matrix functions and analytic continuation, Die Grundlehren der mathematischen Wissenschaften, Band 207, Springer-Verlag, New York-Heidelberg, 1974. MR 0486556
- 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
- Stamatis Koumandos and Henrik L. Pedersen, Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler’s gamma function, J. Math. Anal. Appl. 355 (2009), no. 1, 33–40. MR 2514449, DOI 10.1016/j.jmaa.2009.01.042
- C. Mortici, Monotonicity properties of the volume of the unit ball in $\mathbb R^n$, Optim. Lett. 4 (2010), 457–464.
- Feng Qi, Bai-Ni Guo, Monotonicity and logarithmic convexity relating to the volume of the unit ball, arXiv:0902.2509v1[math.CA].
- Feng Qi, Bai-Ni Guo, and Chao-Ping Chen, Some completely monotonic functions involving the gamma and polygamma functions, J. Aust. Math. Soc. 80 (2006), no. 1, 81–88. MR 2212317, DOI 10.1017/S1446788700011393
- S.-L. Qiu and M. Vuorinen, Some properties of the gamma and psi functions, with applications, Math. Comp. 74 (2005), no. 250, 723–742. MR 2114645, DOI 10.1090/S0025-5718-04-01675-8
- Thomas Jan Stieltjes, Œuvres complètes/Collected papers. Vol. I, II, Springer-Verlag, Berlin, 1993. Reprint of the 1914–1918 edition; Edited and with a preface and a biographical note by Gerrit van Dijk; With additional biographical and historical material by Walter Van Assche, Frits Beukers, Wilhelmus A. J. Luxemburg and Herman J. J. te Riele. MR 1272017
- David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
Additional Information
- Christian Berg
- Affiliation: Institute of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 København Ø, Denmark
- Email: berg@math.ku.dk
- Henrik L. Pedersen
- Affiliation: Department of Basic Sciences and Environment, Faculty of Life Sciences, University of Copenhagen, Thorvaldsensvej 40, DK-1871 Frederiksberg C, Denmark
- Email: henrikp@dina.kvl.dk
- Received by editor(s): December 10, 2009
- Received by editor(s) in revised form: June 9, 2010
- Published electronically: November 19, 2010
- Additional Notes: Both authors acknowledge support by grant 272-07-0321 from the Danish Research Council for Nature and Universe.
- Communicated by: Walter Van Assche
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 139 (2011), 2121-2132
- MSC (2010): Primary 33B15; Secondary 30E20, 30E15
- DOI: https://doi.org/10.1090/S0002-9939-2010-10636-6
- MathSciNet review: 2775390