Inequalities involving the multiple psi function. (Inégalités mettant en jeu la fonction psi multiple.) (English. French summary) Zbl 1387.33002
Author’s abstract: 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.
Reviewer: Yuri A. Farkov (Moscow)
MSC:
| 33B15 | Gamma, beta and polygamma functions |
References:
| [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 (1964), U.S. Government Printing Office: U.S. Government Printing Office Washington, DC, For sale by the Superintendent of Documents · Zbl 0171.38503 |
| [2] | Adamchik, V. S., The multiple gamma function and its application to computation of series, Ramanujan J., 9, 3, 271-288 (2005) · Zbl 1088.33014 |
| [3] | Alzer, H., Mean-value inequalities for the polygamma functions, Aequ. Math., 61, 1-2, 151-161 (2001) · Zbl 0968.33003 |
| [4] | Alzer, H., Sharp inequalities for the digamma and polygamma functions, Forum Math., 16, 2, 181-221 (2004) · Zbl 1048.33001 |
| [5] | Alzer, H., Sub- and superadditive properties of Euler’s gamma function, Proc. Amer. Math. Soc., 135, 11, 3641-3648 (2007), (electronic) · Zbl 1126.33001 |
| [6] | Alzer, H.; Ruehr, O. G., A submultiplicative property of the psi function, J. Comput. Appl. Math., 101, 1-2, 53-60 (1999) · Zbl 0943.33001 |
| [7] | Alzer, H.; Ruscheweyh, S., A subadditive property of the gamma function, J. Math. Anal. Appl., 285, 2, 564-577 (2003) · Zbl 1129.33300 |
| [8] | Barnes, E. W., The theory of the G-function, Q. J. Math., 31, 264-314 (1899) · JFM 30.0389.02 |
| [9] | Barnes, E. W., On the theory of the multiple gamma function, Trans. Camb. Philos. Soc., 19, 374-439 (1904) · JFM 35.0462.01 |
| [10] | Batir, N., Inequalities for the double gamma function, J. Math. Anal. Appl., 351, 1, 182-185 (2009) · Zbl 1157.33001 |
| [11] | Batir, N., Monotonicity properties of \(q\)-digamma and \(q\)-trigamma functions, J. Approx. Theory, 192, 336-346 (2015) · Zbl 1312.33004 |
| [12] | Chen, C.-P., Inequalities associated with Barnes \(G\)-function, Expo. Math., 29, 1, 119-125 (2011) · Zbl 1216.33004 |
| [13] | Choi, J., Determinant of Laplacian on \(S^3\), Math. Jpn., 40, 1, 155-166 (1994) · Zbl 0806.58053 |
| [14] | Choi, J., Determinants of the Laplacians on the \(n\)-dimensional unit sphere \(S^n\), Adv. Differ. Equ., 2013, Article 236 pp. (2013) · Zbl 1375.11061 |
| [15] | Choi, J., Multiple gamma functions and their applications, (Analytic Number Theory, Approximation Theory, and Special Functions (2014), Springer: Springer New York), 93-129 · Zbl 1323.33002 |
| [16] | Choi, J.; Srivastava, H. M., An application of the theory of the double gamma function, Kyushu J. Math., 53, 1, 209-222 (1999) · Zbl 1013.11053 |
| [17] | Choi, J.; Srivastava, H. M., Certain classes of series associated with the zeta function and multiple gamma functions, J. Comput. Appl. Math., 118, 1-2, 87-109 (2000) · Zbl 0969.11030 |
| [18] | Choi, J.; Srivastava, H. M., Some two-sided inequalities for multiple gamma functions and related results, Appl. Math. Comput., 219, 20, 10343-10354 (2013) · Zbl 1293.33001 |
| [19] | Shabani, A. Sh., Some inequalities for the gamma function, J. Inequal. Pure Appl. Math., 8, 2, Article 49 pp. (2007) · Zbl 1168.33302 |
| [20] | Srivastava, H. M.; Choi, J., Zeta and \(q\)-Zeta Functions and Associated Series and Integrals (2012), Elsevier, Inc.: Elsevier, Inc. Amsterdam · Zbl 1239.33002 |
| [21] | Ueno, K.; Nishizawa, M., The multiple gamma function and its \(q\)-analogue, (Quantum Groups and Quantum Spaces. Quantum Groups and Quantum Spaces, Warsaw, 1995. Quantum Groups and Quantum Spaces. Quantum Groups and Quantum Spaces, Warsaw, 1995, Banach Cent. Publ., vol. 40 (1997), Polish Acad. Sci.: Polish Acad. Sci. Warsaw), 429-441 · Zbl 0869.33001 |
| [22] | Vignéras, M. F., L’équation fonctionnelle de la fonction zeta de Selberg de groupe modulaire PSL(2; Z), Astérisque, 61, 235-249 (1979) · Zbl 0401.10036 |
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