Abstract
For entire or meromorphic function f, a value θ ∈ [0, 2π) is called a Julia limiting direction if there is an unbounded sequence {zn} in the Julia set satisfying \(\mathop {\lim }\limits_{n \to \infty } \;\arg {z_n} = \theta \). Our main result is on the entire solution f of P(z, f) + F(z)fs = 0, where P(z, f) is a differential polynomial of f with entire coefficients of growth smaller than that of the entire transcendental F, with the integer s being no more than the minimum degree of all differential monomials in P(z, f). We observe that Julia limiting directions of f partly come from the directions in which F grows quickly.
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References
Albert B. Proof of Edrei’s Spread Conjecture. Proceedings of the London Mathematical Society, 1973, 26(3): 418–434
Baker I N. Sets of Non-Normality in Iteration Theory. Journal of the London Mathematical Society, 1965, 40(1): 499–502
Baker I N. The domains of normality of an entire function. Annales Academiae entiarum Fennicae Series A I Mathematica, 1975, 1(2): 277–283
Bergweiler, Walter. Iteration of meromorphic functions. Bulletin of the American Mathematical Society, 1993, 29(2): 151–188
Edrei A. Sums of deficiencies of meromorphic functions. Journal d Analyse Mathematique, 1965, 14(1): 79–107
Eremenko A E. On the iteration of entire functions. Banach Center Publications, 1989, 23(1): 339–345
Goldberg A A, Ostrowskii I V. Value distribution of meromorphic functions. AMS Translations of Mathematical Monographs series, 2008
Hayman W K. Meromorphic Functions. London: Oxford University Press, 1964
Huang Z G, Wang J. On limit directions of Julia sets of entire solutions of linear differential equations. Journal of Mathematical Analysis & Applications, 2014, 409(1): 478–484
Laine I. Nevanlinna theory and complex differential equations. Berlin: de Gruyter, 1993
Qiao J. On limiting directions of Julia sets. Annales Academiae Scientiarum Fennicae Mathematica, 2001, 26(2): 391–399
Qiu L, Wu S. Radial distributions of Julia sets of meromorphic functions. Journal of the Australian Mathematical Society, 2006, 81(3): 363–368
Rudin W. Real and complex analysis. Osborne McGraw-Hill, 1974
Schleicher D. Dynamics of Entire Functions//Holomorphic Dynamical Systems. Berlin Heidelberg: Springer Press, 2010: 295–339
Hans Töpfer. Über die Iteration der ganzen transzendenten Funktionen, insbesondere von sin z und cos z. Mathematische Annalen, 1940, 117(1): 65–84
Wang J, Chen Z X. Limiting direction and baker wandering domain of entire solutions of differential equations. Acta Mathematica Scientia, 2016, 36(5): 1331–1342
Wang J, Chen Z X. Limiting directions of Julia sets of entire solutions to complex differential equations. Acta Mathematica Scientia, 2017, 37(1): 97–107
Yang C C, Li P. On the transcendental solutions of a certain type of nonlinear differential equations. Archiv der Mathematik, 2004, 82(5): 442–448
Yang L. Value distribution theory. Springer Science & Business Media, 2013
Zhang G, Yang L. On radial distributions of Julia sets of Newton’s method of solutions of complex differential equations. Proceedings of the Japan Academy Series A-Mathematical Sciences, 2016, 92(1): 1–6
Zheng J H, Wang S, Huang Z G. Some properties of Fatou and Julia sets of transcendental meromorphic functions. Bulletin of the Australian Mathematical Society, 2002, 66: 1–8
Zheng J H. Value Distribution of Meromorphic Functions. TUP-Springer Project, Tsinghua University Press and Springer Press, 2010
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This work was supported by the National Natural Science Foundation of China (11771090, 11901311) and Natural Sciences Foundation of Shanghai (17ZR1402900).
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Wang, J., Yao, X. & Zhang, C. Julia Limiting Directions of Entire Solutions of Complex Differential Equations. Acta Math Sci 41, 1275–1286 (2021). https://doi.org/10.1007/s10473-021-0415-7
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DOI: https://doi.org/10.1007/s10473-021-0415-7