Universal convexity and range problems of shifted hypergeometric functions
HTML articles powered by AMS MathViewer
- by Toshiyuki Sugawa, Li-Mei Wang and Chengfa Wu;
- Proc. Amer. Math. Soc. 152 (2024), 3521-3535
- DOI: https://doi.org/10.1090/proc/16849
- Published electronically: June 18, 2024
- HTML | PDF | Request permission
Abstract:
In the present paper, we study the shifted hypergeometric function $f(z)=z_{2}F_{1}(a,b;c;z)$ for real parameters with $0<a\le b\le c$ and its variant $g(z)=z_{2}F_{2}(a,b;c;z^2)$. Our first purpose is to solve the range problems for $f$ and $g$ posed by Ponnusamy and Vuorinen [Rocky Mountain J. Math. 31 (2001), pp. 327–353]. Ruscheweyh, Salinas and Sugawa [Israel J. Math. 171 (2009), pp. 285–304] developed the theory of universal prestarlike functions on the slit domain $\mathbb {C}\setminus [1,+\infty )$ and showed universal starlikeness of $f$ under some assumptions on the parameters. However, there has been no systematic study of universal convexity of the shifted hypergeometric functions except for the case $b=1$. Our second purpose is to show universal convexity of $f$ under certain conditions on the parameters.References
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1966. MR 208797
- Peter Hästö, S. Ponnusamy, and M. Vuorinen, Starlikeness of the Gaussian hypergeometric functions, Complex Var. Elliptic Equ. 55 (2010), no. 1-3, 173–184. MR 2599619, DOI 10.1080/17476930903276134
- Reinhold Küstner, Mapping properties of hypergeometric functions and convolutions of starlike or convex functions of order $\alpha$, Comput. Methods Funct. Theory 2 (2002), no. 2, [On table of contents: 2004], 597–610. MR 2038140, DOI 10.1007/BF03321867
- Reinhold Küstner, On the order of starlikeness of the shifted Gauss hypergeometric function, J. Math. Anal. Appl. 334 (2007), no. 2, 1363–1385. MR 2338668, DOI 10.1016/j.jmaa.2007.01.011
- Jian-Guo Liu and Robert L. Pego, On generating functions of Hausdorff moment sequences, Trans. Amer. Math. Soc. 368 (2016), no. 12, 8499–8518. MR 3551579, DOI 10.1090/tran/6618
- Bo-Yong Long, Toshiyuki Sugawa, and Qi-Han Wang, Completely monotone sequences and harmonic mappings, Ann. Fenn. Math. 47 (2022), no. 1, 237–250 (English, with English and Finnish summaries). MR 4359800, DOI 10.54330/afm.113314
- Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248
- S. Ponnusamy, Close-to-convexity properties of Gaussian hypergeometric functions, J. Comput. Appl. Math. 88 (1998), no. 2, 327–337. MR 1613250, DOI 10.1016/S0377-0427(97)00221-5
- S. Ponnusamy and M. Vuorinen, Asymptotic expansions and inequalities for hypergeometric functions, Mathematika 44 (1997), no. 2, 278–301. MR 1600537, DOI 10.1112/S0025579300012602
- S. Ponnusamy and M. Vuorinen, Univalence and convexity properties for Gaussian hypergeometric functions, Rocky Mountain J. Math. 31 (2001), no. 1, 327–353. MR 1821384, DOI 10.1216/rmjm/1008959684
- Stephan Ruscheweyh, Convolutions in geometric function theory, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 83, Presses de l’Université de Montréal, Montreal, QC, 1982. Fundamental Theories of Physics. MR 674296
- Stephan Ruscheweyh, Luis Salinas, and Toshiyuki Sugawa, Completely monotone sequences and universally prestarlike functions, Israel J. Math. 171 (2009), 285–304. MR 2520111, DOI 10.1007/s11856-009-0050-9
- Toshiyuki Sugawa and Li-Mei Wang, Notes on convex functions of order $\alpha$, Comput. Methods Funct. Theory 16 (2016), no. 1, 79–92. MR 3460542, DOI 10.1007/s40315-015-0122-2
- Toshio Umezawa, Analytic functions convex in one direction, J. Math. Soc. Japan 4 (1952), 194–202. MR 51313, DOI 10.2969/jmsj/00420194
- H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Co., Inc., New York, 1948. MR 25596
- Li-Mei Wang, On the order of convexity for the shifted hypergeometric functions, Comput. Methods Funct. Theory 21 (2021), no. 3, 505–522. MR 4299911, DOI 10.1007/s40315-021-00383-8
- Li-Mei Wang, Mapping properties of the zero-balanced hypergeometric functions, J. Math. Anal. Appl. 505 (2022), no. 1, Paper No. 125448, 13. MR 4280843, DOI 10.1016/j.jmaa.2021.125448
- Li-Mei Wang, Corrigendum to “Mapping properties of the zero-balanced hypergeometric functions” [J. Math. Anal. Appl. 505 (2022) 125448], J. Math. Anal. Appl. 532 (2024), no. 1, Paper No. 127988, 2. MR 4677705, DOI 10.1016/j.jmaa.2023.127988
Bibliographic Information
- Toshiyuki Sugawa
- Affiliation: Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan
- MR Author ID: 318760
- ORCID: 0000-0002-3429-5498
- Email: sugawa@math.is.tohoku.ac.jp
- Li-Mei Wang
- Affiliation: School of Statistics, University of International Business and Economics, No. 10, Huixin Dongjie, Chaoyang District, Beijing 100029, People’s Republic of China
- Email: wangmabel@163.com
- Chengfa Wu
- Affiliation: Institute for Advanced Study, Shenzhen University, Shenzhen 518060, People’s Republic of China; and School of Mathematical Sciences, Shenzhen University, Shenzhen 518060, People’s Republic of China
- MR Author ID: 1042219
- ORCID: 0000-0003-1697-4654
- Email: cfwu@szu.edu.cn
- Received by editor(s): September 6, 2023
- Received by editor(s) in revised form: February 14, 2024
- Published electronically: June 18, 2024
- Additional Notes: The second author was supported by a grant of University of International Business and Economics (No. 78210418) and National Natural Science Foundation of China (No. 11901086).
The third author was supported by Shenzhen Natural Science Fund (Stable Support Project of Shenzhen, Grant No. 20231121103530003). - Communicated by: Filippo Bracci
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3521-3535
- MSC (2020): Primary 30C45; Secondary 33C05
- DOI: https://doi.org/10.1090/proc/16849
- MathSciNet review: 4767281