Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T11:02:42.984Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Horizontal Monotonicity of |Γ(s)|

Published online by Cambridge University Press:  20 November 2018

Gopala Krishna Srinivasan  and
P. Zvengrowski
Gopala Krishna Srinivasan
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India e-mail: [email protected]
P. Zvengrowski
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4 e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Writing $s\,=\,\sigma \,+\,it$ for a complex variable, it is proved that the modulus of the gamma function, $\left| \Gamma (s) \right|$, is strictly monotone increasing with respect to $\sigma $ whenever $\left| t \right|\,>\,5/4$. It is also shown that this result is false for $\left| t \right|\,\le \,1$.

Keywords

33B15Gamma functionmodulusmonotonicity
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 3 , 01 September 2011 , pp. 538 - 543
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Alzer, H., On some inequalities for the gamma and psi functions. Math. Comp. 66(1997), no. 217, 373–389. doi:10.1090/S0025-5718-97-00807-7Google Scholar
[2] Alzer, H., Monotonicity properties of the Hurwitz zeta function. Canad. Math. Bull. 48(2005), no. 3, 333–339.Google Scholar
[3] Andrews, G. E., Askey, R., and Roy, R., Special functions. Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999.Google Scholar
[4] Davis, P. J., Leonhard Euler's integral: A historical profile of the gamma function. Amer. Math. Monthly 66(1959), 849–869. doi:10.2307/2309786Google Scholar
[5] Edwards, H. M., Riemann's zeta function. Pure and Applied Mathematics, 58, Academic Press, New York-London, 1974.Google Scholar
[6] Euler, L., De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt. In: Opera Omnia Series 1, 14, B. G. Teubner, Berlin, 1925, pp. 1–24.Google Scholar
[7] Gauss, C. F., Disquisitiones generales circa seriem infinitam . In: Werke, 3, Königlichen Gesellschaft der Wissenschaften, Göttingen, 1866.Google Scholar
[8] Godefroy, M., La fonction gamma: théorie, histoire, bibliographie. Gauthier-Villars, Paris, 1901.Google Scholar
[9] Jahnke, E. and Emde, F., Tables of functions with formulae and curves. 4th Ed., Dover, New York, 1945.Google Scholar
[10] Landau, E., Handbuch der Lehre von der Verteilung der Primzahlen. 2nd ed., Chelsea Publishing Co., New York, 1953.Google Scholar
[11] Remmert, R., Classical topics in complex function theory. Graduate Texts in Mathematics, 172, Springer-Verlag, New York, 1998.Google Scholar
[12] Saidak, F. and Zvengrowski, P., On the modulus of the Riemann zeta function in the critical strip. Math. Slovaca 53(2003), no. 2, 145–172.Google Scholar
[13] Srinivasan, G. K., The gamma function: an eclectic tour. Amer. Math. Monthly 114(2007), no. 4, 297–315.Google Scholar
[14] Stieltjes, T.-J., Sur le développement de log Γ(a) . J. Math. Pures Appl. (9) 5(1889), 425–444.Google Scholar
[15] Whittaker, E. T. and Watson, G. N., 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, Cambridge, 1996.Google Scholar