Complex analysis
Inequalities involving the multiple psi function
Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 288-292.

In this work, multiple gamma functions of order n have been considered. The logarithmic derivative of the multiple gamma function is known as the multiple psi function. Subadditive, superadditive, and convexity properties of higher-order derivatives of the multiple psi function are derived. Some related inequalities for these functions and their ratios are also obtained.

Nous considérons ici les fonctions gamma multiples d'ordre n. La dérivée logarithmique de la fonction gamma multiple est la fonction psi bien connue. Nous obtenons des propriétés additives et de convexité des dérivées d'ordre supérieur de la fonction psi multiple. Nous obtenons également quelques inégalités faisant intervenir ces fonctions et leurs quotients.

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DOI: 10.1016/j.crma.2018.01.014
Das, Sourav 1

1 Department of Mathematics, National Institute of Technology, Hamirpur, Himachal Pradesh, 177005, India
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Das, Sourav. Inequalities involving the multiple psi function. Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 288-292. doi : 10.1016/j.crma.2018.01.014. http://www.numdam.org/articles/10.1016/j.crma.2018.01.014/

[1] Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, vol. 55, U.S. Government Printing Office, Washington, DC, 1964 (For sale by the Superintendent of Documents)

[2] Adamchik, V.S. The multiple gamma function and its application to computation of series, Ramanujan J., Volume 9 (2005) no. 3, pp. 271-288

[3] Alzer, H. Mean-value inequalities for the polygamma functions, Aequ. Math., Volume 61 (2001) no. 1–2, pp. 151-161

[4] Alzer, H. Sharp inequalities for the digamma and polygamma functions, Forum Math., Volume 16 (2004) no. 2, pp. 181-221

[5] Alzer, H. Sub- and superadditive properties of Euler's gamma function, Proc. Amer. Math. Soc., Volume 135 (2007) no. 11, pp. 3641-3648 (electronic)

[6] Alzer, H.; Ruehr, O.G. A submultiplicative property of the psi function, J. Comput. Appl. Math., Volume 101 (1999) no. 1–2, pp. 53-60

[7] Alzer, H.; Ruscheweyh, S. A subadditive property of the gamma function, J. Math. Anal. Appl., Volume 285 (2003) no. 2, pp. 564-577

[8] Barnes, E.W. The theory of the G-function, Q. J. Math., Volume 31 (1899), pp. 264-314

[9] Barnes, E.W. On the theory of the multiple gamma function, Trans. Camb. Philos. Soc., Volume 19 (1904), pp. 374-439

[10] Batir, N. Inequalities for the double gamma function, J. Math. Anal. Appl., Volume 351 (2009) no. 1, pp. 182-185

[11] Batir, N. Monotonicity properties of q-digamma and q-trigamma functions, J. Approx. Theory, Volume 192 (2015), pp. 336-346

[12] Chen, C.-P. Inequalities associated with Barnes G-function, Expo. Math., Volume 29 (2011) no. 1, pp. 119-125

[13] Choi, J. Determinant of Laplacian on S3, Math. Jpn., Volume 40 (1994) no. 1, pp. 155-166

[14] Choi, J. Determinants of the Laplacians on the n-dimensional unit sphere Sn, Adv. Differ. Equ., Volume 2013 (2013)

[15] Choi, J. Multiple gamma functions and their applications, Analytic Number Theory, Approximation Theory, and Special Functions, Springer, New York, 2014, pp. 93-129

[16] Choi, J.; Srivastava, H.M. An application of the theory of the double gamma function, Kyushu J. Math., Volume 53 (1999) no. 1, pp. 209-222

[17] Choi, J.; Srivastava, H.M. Certain classes of series associated with the zeta function and multiple gamma functions, J. Comput. Appl. Math., Volume 118 (2000) no. 1–2, pp. 87-109

[18] Choi, J.; Srivastava, H.M. Some two-sided inequalities for multiple gamma functions and related results, Appl. Math. Comput., Volume 219 (2013) no. 20, pp. 10343-10354

[19] Shabani, A.Sh. Some inequalities for the gamma function, J. Inequal. Pure Appl. Math., Volume 8 (2007) no. 2

[20] Srivastava, H.M.; Choi, J. Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, Inc., Amsterdam, 2012

[21] Ueno, K.; Nishizawa, M. The multiple gamma function and its q-analogue, Warsaw, 1995 (Banach Cent. Publ.), Volume vol. 40, Polish Acad. Sci., Warsaw (1997), pp. 429-441

[22] Vignéras, M.F. L'équation fonctionnelle de la fonction zeta de Selberg de groupe modulaire PSL(2; Z), Astérisque, Volume 61 (1979), pp. 235-249

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