Articles

LIMITING DIRECTIONS OF JULIA SETS OF ENTIRE SOLUTIONS TO COMPLEX DIFFERENTIAL EQUATIONS

  • Jun WANG ,
  • Zongxuan CHEN
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  • 1. School of Mathematical Sciences, Fudan University, Shanghai 200433, China;
    2. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
Jun WANG,E-mail:majwang@fudan.edu.cn

Received date: 2015-11-04

  Online published: 2017-02-25

Supported by

The work was supported by Shanghai Center for Mathematical Sciences, China Scholarship Council (201206105015), National Science Foundation of China (11171119, 11001057, 11571049) and Natural Science Foundation of Guangdong Province in China (2014A030313422).

Abstract

In this paper, we mainly investigate the dynamical properties of entire solutions of complex differential equations. With some conditions on coefficients, we prove that the set of common limiting directions of Julia sets of solutions, their derivatives and their primitives must have a definite range of measure.

Cite this article

Jun WANG , Zongxuan CHEN . LIMITING DIRECTIONS OF JULIA SETS OF ENTIRE SOLUTIONS TO COMPLEX DIFFERENTIAL EQUATIONS[J]. Acta mathematica scientia, Series B, 2017 , 37(1) : 97 -107 . DOI: 10.1016/S0252-9602(16)30118-7

References

[1] Baernstein A. Proof of Edrei's spread conjecture. Proc London Math Soc, 1973, 26:418-434
[2] Baker I N. Sets of non-normality in iteration theory. J London Math Soc, 1965, 40:499-502
[3] Baker I N. The domains of normality of an entire function. Ann Acad Sci Fs A I M, 1975, 1:177-183
[4] Bergweiler W. Iteration of meromorphic functions. Bull Amer Math Soc, 1993, 29:151-188
[5] Edrei A. Sums of deficiencies of meromorphic functions. J Anal Math, 1965, 14:79-107
[6] Edrei A. Sums of deficiencies of meromorphic functions II. J Anal Math, 1967, 19:53-74
[7] Fuchs W H J. Proof of a conjecture of G. Pólya concerning gap series. Illinois J Math, 1963, 7:661-667
[8] Goldberg A A, Ostrovskii I V. Value Distribution of Meromorphic Functions. Translations of Mathematical Monographs Series. Providence RI:Amer Math Soc, 2008
[9] Gundersen G. Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J London Math Soc, 1988, 37(2):88-104
[10] Hayman W K. Meromorphic Functions. Oxford:Clarendon Press, 1964
[11] Hayman W K, Rossi J F. Characteristic, maximum modulus and value distribution. Trans Amer Math Soc, 1984, 284:651-664
[12] Huang Z G, Wang J. On the radial distribution of Julia sets of entire solutions of f(k)+A(z)f=0. J Math Anal Appl, 2012, 387:1106-1113
[13] Huang Z G, Wang J. On limit directions of Julia sets of entire solutions of linear differential equations. J Math Anal Appl, 2014, 409:478-484
[14] Laine I. Nevanlinna Theory and Complex Differential Equations. Berlin:Walter de Gruyter, 1993
[15] Lan S T, Chen Z X. On zeros of solutions of higher order homogeneous linear differential equations. Acta Math Sci, 2013, 33B(2):556-564
[16] Murai T. The deficiency of entire functions with Fejér gaps. Ann Inst Fourier (Grenoble), 1983, 33:39-58
[17] Polyanin A D, Zaitsev V F. Handbook of Exact Solutions for Ordinary Differential Equations. 2nd ed. Boca Raton:Chapman and Hall/CRC Press, 2003
[18] Qiao J Y. The value distribution and the Julia sets of entire functions (Chinese). Acta Math Sinica, 1993, 36:418-422
[19] Qiao J Y. Stable domains in the iteration of entire functions (Chinese). Acta Math Sinica, 1994, 37:702-708
[20] Qiao J Y. On limiting directions of Julia sets. Ann Acad Sci Fenn Math, 2001, 26:391-399
[21] Qiu L, Wu S J. Radial distributions of Julia sets of meromorphic functions. J Austral Math Soc, 2006, 81(3):363-368
[22] Wang S P. On the sectorial oscillation theory of f"+Af=0. Ann Acad Sci Fenn Ser A I M Disser, 1994, 92:1-60
[23] Yang C C, Yi H X. Uniqueness Theory of Meromorphic Functions. Dordrecht:Kluwer Press, 2003
[24] Yang L. New Researches on Value Distribution (Chinese). Beijing:Science Press of China, 1982
[25] Zhang G W, Ding J, Yang L Z. Radial dsitribution of Julia sets of derivetives of solutions to complex linear differential equations (Chinese). Sci Sin Math, 2014, 44:693-700
[26] Zhang G W, Wang J, Yang L Z. On radial distribution of Julia sets of solutions to certain second order complex differential equations. Abs Appl Anal, 2014, 2014:Article ID 842693
[27] Zheng J H, Wang S, Huang Z G. Some properties of Fatou and Julia sets of transcendental meromorphic functions. Bull Austral Math Soc, 2002, 66:1-8
[28] Zheng J H. Value Dictribution of Meromorphic Functions. Beijing:Tsinghua University Press, Springer Press, 2010
[29] Zheng J H. Dynamics of Transcendental Meromorphic Functions (Chinese). Monograph of Tsinghua University. Beijing:Tsinghua University Press, 2006

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