Geodesic
On the Horizontal Monotonicity of |Γ(s)|
Canadian mathematical bulletin, Tome 54 (2011) no. 3, pp. 538-543

Voir la notice de l'article provenant de la source Cambridge University Press

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 $\left| t \right|\,\le \,1$ .
DOI : 10.4153/CMB-2010-107-8
Mots-clés : 33B15, Gamma function, modulus, monotonicity 
Srinivasan, Gopala Krishna; Zvengrowski, P. On the Horizontal Monotonicity of |Γ(s)|. Canadian mathematical bulletin, Tome 54 (2011) no. 3, pp. 538-543. doi: 10.4153/CMB-2010-107-8
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[1] [1] Alzer, H., On some inequalities for the gamma and psi functions. Math. Comp. 66(1997), no. 217, 373–389. doi:10.1090/S0025-5718-97-00807-7 Google Scholar

[2] [2] Alzer, H., Monotonicity properties of the Hurwitz zeta function. Canad. Math. Bull. 48(2005), no. 3, 333–339. Google Scholar

[3] [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] [4] Davis, P. J., Leonhard Euler's integral: A historical profile of the gamma function. Amer. Math. Monthly 66(1959), 849–869. doi:10.2307/2309786 Google Scholar

[5] [5] Edwards, H. M., Riemann's zeta function. Pure and Applied Mathematics, 58, Academic Press, New York-London, 1974. Google Scholar

[6] [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] [7] Gauss, C. F., Disquisitiones generales circa seriem infinitam . In: Werke, 3, Königlichen Gesellschaft der Wissenschaften, Göttingen, 1866. Google Scholar

[8] [8] Godefroy, M., La fonction gamma: théorie, histoire, bibliographie. Gauthier-Villars, Paris, 1901. Google Scholar

[9] [9] Jahnke, E. and Emde, F., Tables of functions with formulae and curves. 4th Ed., Dover, New York, 1945. Google Scholar

[10] [10] Landau, E., Handbuch der Lehre von der Verteilung der Primzahlen. 2nd ed., Chelsea Publishing Co., New York, 1953. Google Scholar

[11] [11] Remmert, R., Classical topics in complex function theory. Graduate Texts in Mathematics, 172, Springer-Verlag, New York, 1998. Google Scholar

[12] [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] [13] Srinivasan, G. K., The gamma function: an eclectic tour. Amer. Math. Monthly 114(2007), no. 4, 297–315. Google Scholar

[14] [14] Stieltjes, T.-J., Sur le développement de log Γ(a) . J. Math. Pures Appl. (9) 5(1889), 425–444. Google Scholar

[15] [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

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