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Functions of small growth with no unbounded Fatou components

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Abstract

We prove a form of the cos πρ theorem which gives strong estimates for the minimum modulus of a transcendental entire function of order zero. We also prove a generalisation of a result of Hinkkanen that gives a sufficient condition for a transcendental entire function to have no unbounded Fatou components. These two results enable us to show that there is a large class of entire functions of order zero which have no unbounded Fatou components. On the other hand, we give examples which show that there are in fact functions of order zero which not only fail to satisfy Hinkkanen’s condition but also fail to satisfy our more general condition. We also give a new regularity condition that is sufficient to ensure that a transcendental entire function of order less than 1/2 has no unbounded Fatou components. Finally, we observe that all the conditions given here which guarantee that a transcendental entire function has no unbounded Fatou components also guarantee that the escaping set is connected, thus answering a question of Eremenko for such functions.

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References

  1. J. M. Anderson and A. Hinkkanen, Unbounded domains of normality, Proc. Amer. Math. Soc. 126 (1998), 3243–3252.

    Article  MATH  MathSciNet  Google Scholar 

  2. I. N. Baker, Zusammensetzungen ganzer Funktionen, Math. Z. 69 (1958), 121–163.

    Article  MATH  MathSciNet  Google Scholar 

  3. I. N. Baker, The iteration of polynomials and transcendental entire functions, J. Austral. Math. Soc. Series A 30 (1980/81), 483–495.

    Article  Google Scholar 

  4. P. D. Barry, On a theorem of Besicovitch, Quart. J. Math. Oxford Ser. (2) 14 (1963), 293–302.

  5. W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. 29 (1993), 151–188.

    Article  MATH  MathSciNet  Google Scholar 

  6. W. Bergweiler and A. Hinkkanen, On semiconjugation of entire functions, Math. Proc. Cambridge Philos. Soc. 126 (1999), 565–574.

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Cartan, Sur les systèmes de fonctions holomorphes à variátás linéaires et leurs applications, Ann. Sci. École Norm. Sup. (3) 45 (1928), 255–346.

    Google Scholar 

  8. M. L. Cartwright, Integral Functions, Cambridge University Press, 1962.

  9. A. E. Eremenko, On the iteration of entire functions, in Dynamical Systems and Ergodic Theory, Banach Center Publ. 23, Polish Scientific Publishers, Warsaw, 1989, pp. 339–345.

    Google Scholar 

  10. A. Hinkkanen, Entire functions with no unbounded Fatou components, in Complex Analysis and Dynamical Systems II, Contemp. Math. 382, Amer. Math. Soc., Providence, RI, 2005, pp. 217–226.

    Google Scholar 

  11. A. Hinkkanen, Entire functions with bounded Fatou components, in Transcendental Dynamics and Complex Analysis, Cambridge University Press, 2008, pp. 187–216.

  12. A. Hinkkanen and J. Miles, Growth conditions for entire functions with only bounded Fatou components, J. Anal. Math. 108 (2009), 87–118.

    Article  MATH  MathSciNet  Google Scholar 

  13. X. Hua and C. C. Yang, Fatou components of entire functions of small growth, Ergodic Theory Dynam. Systems 19 (1999), 1281–1293.

    Article  MATH  MathSciNet  Google Scholar 

  14. P. J. Rippon and G. M. Stallard, On sets where iterates of a meromorphic function zip towards infinity, Bull. London Math. Soc. 32 (2000), 528–536.

    Article  MATH  MathSciNet  Google Scholar 

  15. P. J. Rippon and G. M. Stallard, On questions of Fatou and Eremenko, Proc. Amer. Math. Soc. 133 (2005), 1119–1126.

    Article  MATH  MathSciNet  Google Scholar 

  16. P. J. Rippon and G. M. Stallard, Escaping points of entire functions of small growth, Math. Z. 261 (2009), 557–570.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. P. Singh, Composite entire functions with no unbounded Fatou components, J. Math. Anal. Appl. 335 (2007), 907–914.

    Article  MATH  MathSciNet  Google Scholar 

  18. G. M. Stallard, The iteration of entire functions of small growth, Math. Proc. Cambridge Philos. Soc. 114 (1993), 43–55.

    Article  MATH  MathSciNet  Google Scholar 

  19. E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford University Press, 1939.

  20. Y. Wang, Bounded domains of the Fatou set of an entire function, Israel J. Math. 121 (2001), 55–60.

    Article  MATH  MathSciNet  Google Scholar 

  21. Jian-Hua Zheng, Unbounded domains of normality of entire functions of small growth, Math. Proc. Cambridge Philos. Soc. 128 (2000), 355–361.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA, UK

    P. J. Rippon & G. M. Stallard

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  1. P. J. Rippon
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  2. G. M. Stallard
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Correspondence to P. J. Rippon.

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Rippon, P.J., Stallard, G.M. Functions of small growth with no unbounded Fatou components. J Anal Math 108, 61–86 (2009). https://doi.org/10.1007/s11854-009-0018-z

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  • DOI: https://doi.org/10.1007/s11854-009-0018-z

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