Proceedings of the Japan Academy, Series A, Mathematical Sciences

On radial distributions of Julia sets of Newton’s method of solutions of complex differential equations

Guowei Zhang and Lianzhong Yang

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Abstract

In this paper we mainly investigate the radial distribution of Julia sets of Newton’s method of entire solutions of some complex linear differential equations. Under certain conditions, we find the lower bound of it and also obtain some related results.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 1 (2016), 1-6.

Dates
First available in Project Euclid: 28 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.pja/1451330559

Digital Object Identifier
doi:10.3792/pjaa.92.1

Mathematical Reviews number (MathSciNet)
MR3447742

Zentralblatt MATH identifier
1372.30014

Subjects
Primary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX] 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04] 37F50: Small divisors, rotation domains and linearization; Fatou and Julia sets

Keywords
Radial distribution Julia set Newton’s method complex differential equation

Citation

Zhang, Guowei; Yang, Lianzhong. On radial distributions of Julia sets of Newton’s method of solutions of complex differential equations. Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 1, 1--6. doi:10.3792/pjaa.92.1. https://projecteuclid.org/euclid.pja/1451330559


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References

  • A. Baernstein II, Proof of Edrei's spread conjecture, Proc. London Math. Soc. (3) 26 (1973), 418–434.
  • I. N. Baker, Sets of non-normality in iteration theory, J. London Math. Soc. 40 (1965), 499–502.
  • I. N. Baker and P. Domínguez, Boundaries of unbounded Fatou components of entire functions, Ann. Acad. Sci. Fenn. Math. 24 (1999), no. 2, 437–464.
  • K. Barański, N. Fagella, X. Jarque and B. Karpińska, On the connectivity of the Julia sets of meromorphic functions, Invent. Math. 198 (2014), no. 3, 591–636.
  • W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 151–188.
  • R. Brück, On entire functions which share one value CM with their first derivative, Results Math. 30 (1996), no. 1–2, 21–24.
  • A. A. Goldberg and I. V. Ostrovskii, Value distribution of meromorphic functions, translated from the 1970 Russian original by Mikhail Ostrovskii, Translations of Mathematical Monographs, 236, Amer. Math. Soc., Providence, RI, 2008.
  • G. G. Gundersen and L.-Z. Yang, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl. 223 (1998), no. 1, 88–95.
  • W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.
  • Z. Huang and J. Wang, On the radial distribution of Julia sets of entire solutions of $f^{(n)}+A(z)f=0$, J. Math. Anal. Appl. 387 (2012), no. 2, 1106–1113.
  • Z.-G. Huang and J. Wang, On limit directions of Julia sets of entire solutions of linear differential equations, J. Math. Anal. Appl. 409 (2014), no. 1, 478–484.
  • I. Laine, Nevanlinna theory and complex differential equations, de Gruyter Studies in Mathematics, 15, de Gruyter, Berlin, 1993.
  • J. Y. Qiao, Stable sets for iterations of entire functions, Acta Math. Sinica 37 (1994), no. 5, 702–708.
  • J. Qiao, On limiting directions of Julia sets, Ann. Acad. Sci. Fenn. Math. 26 (2001), no. 2, 391–399.
  • L. Qiu and S. Wu, Radial distributions of Julia sets of meromorphic functions, J. Aust. Math. Soc. 81 (2006), no. 3, 363–368.
  • S. Wang, On radial distribution of Julia sets of meromorphic functions, Taiwanese J. Math. 11 (2007), no. 5, 1301–1313.
  • S. P. Wang, On the sectorial oscillation theory of $f''+A(z)f=0$, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 92 (1994), 1–66.
  • H. Wittich, Neuere Untersuchungen über eindeutige analytische Funktionen, Zweite, korrigierte Auflage. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 8, Springer, Berlin, 1968.
  • L. Yang, Value distribution theory, translated and revised from the 1982 Chinese original, Springer, Berlin, 1993.
  • L. Yang, Borel directions of meromorphic functions in an angular domain, Sci. Sinica (1979), Special Issue I on Math., 149–164.
  • C.-C. Yang and H.-X. Yi, Uniqueness theory of meromorphic functions, Mathematics and its Applications, 557, Kluwer Acad. Publ., Dordrecht, 2003.
  • G. W. Zhang, J. Ding and L. Z. Yang, Radial distribution of Julia sets of derivatives of solutions to complex linear differential equations (in Chinese), Sci. Sin. Math. 44 (2014), 693–700.
  • J.-H. Zheng, Dynamics of a meromorphic function (in Chinese), Tsinghua University Press, Beijing, 2006.
  • J.-H. Zheng, S. Wang and Z.-G. Huang, Some properties of Fatou and Julia sets of transcendental meromorphic functions, Bull. Austral. Math. Soc. 66 (2002), no. 1, 1–8.