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Herman rings of meromorphic maps with an omitted value


Author: Tarakanta Nayak
Journal: Proc. Amer. Math. Soc. 144 (2016), 587-597
MSC (2010): Primary 37F10; Secondary 32A20
DOI: https://doi.org/10.1090/proc12715
Published electronically: August 20, 2015
MathSciNet review: 3430836
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Abstract | References | Similar Articles | Additional Information

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.


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  • [1] 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 (94c:58166), https://doi.org/10.1007/BF03323112
  • [2] 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 (99m:30051)
  • [3] 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 (96h:30055), https://doi.org/10.4171/RMI/176
  • [4] A. Bolsch, Periodic Fatou components of meromorphic functions, Bull. London Math. Soc. 31 (1999), no. 5, 543-555. MR 1703869 (2000e:30046), https://doi.org/10.1112/S0024609399005950
  • [5] Patricia Domínguez and Núria Fagella, Existence of Herman rings for meromorphic functions, Complex Var. Theory Appl. 49 (2004), no. 12, 851-870. MR 2101211 (2005h:37096), https://doi.org/10.1080/02781070412331298589
  • [6] 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, https://doi.org/10.1016/j.jmaa.2012.05.005
  • [7] 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, https://doi.org/10.1112/jlms/jdq065
  • [8] 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 (97f:30033)
  • [9] Jian-Hua Zheng, Remarks on Herman rings of transcendental meromorphic functions, Indian J. Pure Appl. Math. 31 (2000), no. 7, 747-751. MR 1779436 (2001i:37068)

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Additional Information

Tarakanta Nayak
Affiliation: School of Basic Sciences, Indian Institute of Technology, Bhubaneswar, India
Email: tnayak@iitbbs.ac.in

DOI: https://doi.org/10.1090/proc12715
Keywords: Herman ring, omitted value, meromorphic function
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
Article copyright: © Copyright 2015 American Mathematical Society

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