Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE INFINITE GROWTH OF SOLUTIONS OF SECOND ORDER LINEAR COMPLEX DIFFERENTIAL EQUATIONS WITH COMPLETELY REGULAR GROWTH COEFFICIENT

  • Zhang, Guowei (School of Mathematics and Statistics Anyang Normal University)
  • Received : 2020.04.07
  • Accepted : 2020.07.09
  • Published : 2021.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we discuss the classical problem of finding conditions on the entire coefficients A(z) and B(z) guaranteeing that all nontrivial solutions of f" + A(z)f' + B(z)f = 0 are of infinite order. We assume A(z) is an entire function of completely regular growth and B(z) satisfies three different conditions, then we obtain three results respectively. The three conditions are (1) B(z) has a dynamical property with a multiply connected Fatou component, (2) B(z) satisfies T(r, B) ~ log M(r, B) outside a set of finite logarithmic measure, (3) B(z) is extremal for Denjoy's conjecture.

Keywords

References

  1. L. Ahlfors, Untersuchungen zur Theorie der konformen abbildung und der Theorie der ganzen Funktionen, Acta Soc. Sci. Fenn. 1 (1930), 1-40.
  2. I. N. Baker, Multiply connected domains of normality in iteration theory, Math. Z. 81 (1963), 206-214. https://doi.org/10.1007/BF01111543
  3. I. N. Baker, The domains of normality of an entire function, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 2, 277-283. https://doi.org/10.5186/aasfm.1975.0101
  4. I. N. Baker, An entire function which has wandering domains, J. Austral. Math. Soc. Ser. A 22 (1976), no. 2, 173-176. https://doi.org/10.1017/s1446788700015287
  5. W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 151-188. https://doi.org/10.1090/S0273-0979-1993-00432-4
  6. W. Bergweiler and I. Chyzhykov, Lebesgue measure of escaping sets of entire functions of completely regular growth, J. Lond. Math. Soc. (2) 94 (2016), no. 2, 639-661. https://doi.org/10.1112/jlms/jdw051
  7. Z. Chen, The growth of solutions of f'' + e-zf' + Q(z)f = 0 where the order (Q) = 1, Sci. China Ser. A 45 (2002), no. 3, 290-300.
  8. A. Denjoy, Sur les fonctions enti'eres de genre fini, C. R. Acad. Sci. Paris, 45 (1907), 106-109.
  9. A. E. Eremenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 989-1020. https://doi.org/10.5802/aif.1318
  10. 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, American Mathematical Society, Providence, RI, 2008.
  11. G. G. Gundersen, On the question of whether f'' + e-zf' + B(z)f = 0 can admit a solution f ≢ 0 of finite order, Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), no. 1-2, 9-17. https://doi.org/10.1017/S0308210500014451
  12. G. G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988), no. 1, 88-104. https://doi.org/10.1112/jlms/s2-37.121.88
  13. G. G. Gundersen, Finite order solutions of second order linear differential equations, Trans. Amer. Math. Soc. 305 (1988), no. 1, 415-429. https://doi.org/10.2307/2001061
  14. J. Heittokangas, I. Laine, K. Tohge, and Z. Wen, Completely regular growth solutions of second order complex linear differential equations, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 2, 985-1003. https://doi.org/10.5186/aasfm.2015.4057
  15. S. Hellerstein, J. Miles, and J. Rossi, On the growth of solutions of f'' + gf' + hf = 0, Trans. Amer. Math. Soc. 324 (1991), no. 2, 693-706. https://doi.org/10.2307/2001737
  16. E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley Publ. Co., Reading, MA, 1969.
  17. A. S. B. Holland, Introduction to the Theory of Entire Functions, Academic Press, New York, 1973.
  18. K. Ishizaki and K. Tohge, On the complex oscillation of some linear differential equations, J. Math. Anal. Appl. 206 (1997), no. 2, 503-517. https://doi.org/10.1006/jmaa.1997.5247
  19. I. Laine, Nevanlinna Theory and Complex Differential Equations, De Gruyter Studies in Mathematics, 15, Walter de Gruyter & Co., Berlin, 1993. https://doi.org/10.1515/9783110863147
  20. I. Laine and P. Wu, Growth of solutions of second order linear differential equations, Proc. Amer. Math. Soc. 128 (2000), no. 9, 2693-2703. https://doi.org/10.1090/S0002-9939-00-05350-8
  21. J. K. Langley, On complex oscillation and a problem of Ozawa, Kodai Math. J. 9 (1986), no. 3, 430-439. https://doi.org/10.2996/kmj/1138037272
  22. B. Levin, Distribution of Zeros of Entire Functions, American Mathematical Society, Providence, RI, 1964.
  23. J. R. Long, Growth of solutions of second order complex linear differential equations with entire coefficients, Filomat 32 (2018), no. 1, 275-284. https://doi.org/10.2298/fil1801275l
  24. J. R. Long and K. E. Qiu, Growth of solutions to a second-order complex linear differential equation, Math. Pract. Theory 45 (2015), no. 2, 243-247.
  25. J. R. Long, P. C. Wu, and Z. Zhang, On the growth of solutions of second order linear differential equations with extremal coefficients, Acta Math. Sin. (Engl. Ser.) 29 (2013), no. 2, 365-372. https://doi.org/10.1007/s10114-012-0648-4
  26. T. Murai, The deficiency of entire functions with Fejer gaps, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 3, 39-58. https://doi.org/10.5802/aif.930
  27. E. Schwengeler, Geometrisches uber die Verteilung der Nullstellen spezieller ganzer Funktionen (Exponentialsummen), Dissertation. ETH Zurich, 1925.
  28. D. J. Sixsmith, Julia and escaping set spiders' webs of positive area, Int. Math. Res. Not. IMRN 2015, no. 19, 9751-9774. https://doi.org/10.1093/imrn/rnu245
  29. J. Wang and Z. Chen, Limiting directions of Julia sets of entire solutions to complex differential equations, Acta Math. Sci. Ser. B (Engl. Ed.) 37 (2017), no. 1, 97-107. https://doi.org/10.1016/S0252-9602(16)30118-7
  30. Z.-T. Wen, G. G. Gundersen, and J. Heittokangas, Dual exponential polynomials and linear differential equations, J. Differential Equations 264 (2018), no. 1, 98-114. https://doi.org/10.1016/j.jde.2017.09.003
  31. X. Wu, J. Long, J. Heittokangas, and K. Qiu, Second-order complex linear differential equations with special functions or extremal functions as coefficients, Electron. J. Differential Equations 2015 (2015), No. 143, 15 pp.
  32. X. B. Wu and P. C. Wu, Growth of solutions to the equation f'' + Af' + Bf = 0, where A is a solution to a second-order linear differential equation, Acta Math. Sci. Ser. A (Chin. Ed.) 33 (2013), no. 1, 46-52.
  33. P. Wu and J. Zhu, On the growth of solutions to the complex differential equation f'' + Af' + Bf = 0, Sci. China Math. 54 (2011), no. 5, 939-947. https://doi.org/10.1007/s11425-010-4153-x
  34. C.-C. Yang and H.-X. Yi, Uniqueness theory of meromorphic functions, Mathematics and its Applications, 557, Kluwer Academic Publishers Group, Dordrecht, 2003. https://doi.org/10.1007/978-94-017-3626-8
  35. L. Yang and G. H. Zhang, Distribution of Borel directions of entire functions, Acta Math. Sinica 19 (1976), no. 3, 157-168.
  36. G. H. Zhang, Theory of Entire and Meromorphic Functions-Deficient Values, Asymptotic Values and Singular Directions, Springer-Verlag, Berlin, 1993.
  37. G. Zhang and J. Wang, The infinite growth of solutions of complex differential equations of which coefficient with dynamical property, Taiwanese J. Math. 18 (2014), no. 4, 1063-1069. https://doi.org/10.11650/tjm.18.2014.3902
  38. J.-H. Zheng, On multiply-connected Fatou components in iteration of meromorphic functions, J. Math. Anal. Appl. 313 (2006), no. 1, 24-37. https://doi.org/10.1016/j.jmaa.2005.05.038