Herman rings of meromorphic maps with an omitted value
HTML articles powered by AMS MathViewer
- by Tarakanta Nayak PDF
- Proc. Amer. Math. Soc. 144 (2016), 587-597 Request permission
Abstract:
We investigate the existence and distribution of Herman rings of transcendental meromorphic functions which have at least one omitted value. If all the poles of such a function are multiple, then it has no Herman ring. Herman rings of period one or two do not exist. Functions with a single pole or with at least two poles, one of which is an omitted value, have no Herman ring. Every doubly connected periodic Fatou component is a Herman ring.References
- I. N. Baker, J. Kotus, and Lü Yinian, Iterates of meromorphic functions. IV. Critically finite functions, Results Math. 22 (1992), no. 3-4, 651–656. MR 1189754, DOI 10.1007/BF03323112
- I. N. Baker and P. Domínguez, Analytic self-maps of the punctured plane, Complex Variables Theory Appl. 37 (1998), no. 1-4, 67–91. MR 1687848, DOI 10.1080/17476939808815123
- Walter Bergweiler and Alexandre Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11 (1995), no. 2, 355–373. MR 1344897, DOI 10.4171/RMI/176
- A. Bolsch, Periodic Fatou components of meromorphic functions, Bull. London Math. Soc. 31 (1999), no. 5, 543–555. MR 1703869, DOI 10.1112/S0024609399005950
- Patricia Domínguez and Núria Fagella, Existence of Herman rings for meromorphic functions, Complex Var. Theory Appl. 49 (2004), no. 12, 851–870. MR 2101211, DOI 10.1080/02781070412331298589
- Núria Fagella and Jörn Peter, On the configuration of Herman rings of meromorphic functions, J. Math. Anal. Appl. 394 (2012), no. 2, 458–467. MR 2927468, DOI 10.1016/j.jmaa.2012.05.005
- Tarakanta Nayak and Jian-Hua Zheng, Omitted values and dynamics of meromorphic functions, J. Lond. Math. Soc. (2) 83 (2011), no. 1, 121–136. MR 2763947, DOI 10.1112/jlms/jdq065
- Zhi-Min Gong, Wei-Yuan Qiu, and Fu-Yao Ren, A negative answer to a problem of Bergweiler, Complex Variables Theory Appl. 30 (1996), no. 4, 315–322. MR 1413161, DOI 10.1080/17476939608814933
- Jian-Hua Zheng, Remarks on Herman rings of transcendental meromorphic functions, Indian J. Pure Appl. Math. 31 (2000), no. 7, 747–751. MR 1779436
Additional Information
- Tarakanta Nayak
- Affiliation: School of Basic Sciences, Indian Institute of Technology, Bhubaneswar, India
- MR Author ID: 806609
- Email: tnayak@iitbbs.ac.in
- Received by editor(s): November 3, 2012
- Received by editor(s) in revised form: May 23, 2014, and December 30, 2014
- Published electronically: August 20, 2015
- Additional Notes: The author was supported by the Department of Science & Technology, Govt. of India through the Fast Track Project (SR/FTP/MS-019/2011).
- Communicated by: Nimish Shah
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 587-597
- MSC (2010): Primary 37F10; Secondary 32A20
- DOI: https://doi.org/10.1090/proc12715
- MathSciNet review: 3430836