Taiwanese Journal of Mathematics

Growth of Solutions of Higher Order Complex Linear Differential Equation

Jianren Long and Xiubi Wu

Full-text: Open access

Abstract

Some new conditions on coefficient functions $A_{i}(z)$, which will guarantee all nontrivial solutions of $f^{(n)} + A_{n-1}(z) f^{(n-1)} + \cdots + A_{0}(z)f = 0$ are of infinite order, are found in this paper. The first condition involves two classes of extremal functions for some inequalities about finite asymptotic values and deficient values. The second condition assumes that a coefficient itself is a nontrivial solution of another differential equation $w'' + P(z)w = 0$, where $P(z)$ is a polynomial.

Article information

Source
Taiwanese J. Math., Volume 21, Number 5 (2017), 961-977.

Dates
Received: 24 April 2016
Revised: 27 September 2016
Accepted: 8 January 2017
First available in Project Euclid: 1 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1501599178

Digital Object Identifier
doi:10.11650/tjm/7950

Mathematical Reviews number (MathSciNet)
MR3707879

Zentralblatt MATH identifier
06871354

Subjects
Primary: 34M10: Oscillation, growth of solutions
Secondary: 30D35: Distribution of values, Nevanlinna theory

Keywords
complex differential equation entire function infinite order Denjoy's conjecture Yang's inequality

Citation

Long, Jianren; Wu, Xiubi. Growth of Solutions of Higher Order Complex Linear Differential Equation. Taiwanese J. Math. 21 (2017), no. 5, 961--977. doi:10.11650/tjm/7950. https://projecteuclid.org/euclid.twjm/1501599178


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References

  • L. V. Ahlfors, Untersuchungen zur Theorie der konformen abbildung und der Theorie der ganzen Funktionen, Acta Soc. Sci. Fenn. 1 (1930), 1–40.
  • P. D. Barry, Some theorems related to the $\cos \pi\rho$ theorem, Proc. London. Math. Soc. (3) 21 (1970), 334–360.
  • T.-B. Cao, K. Liu and J. Wang, On the growth of solutions of complex differential equations with entire coefficients of finite logarithmic order, Math. Rep. (Bucur.) 15(65) (2013), no. 3, 249–269.
  • A. Denjoy, Sur les fonctions entières de genre fini, C. R. Acad. Sci. Paris 45 (1907), 106–109.
  • D. Drasin, On asymptotic curves of functions extremal for Denjoy's conjecture, Proc. London Math. Soc. (3) 26 (1973), 142–166.
  • M. Frei, Über die Lösungen linearer Differentialgleichungen mit ganzen Funktionen als Koeffizienten, Comment. Math. Helv. 35 (1961), 201–222.
  • S. A. Gao, Z. X. Chen and T. W. Chen, The Complex Oscillation Theory of Linear Differential Equations, Huazhong University of Science and Technology press, Wuhan, 1997.
  • G. G. Gundersen, On the real zeros of solutions of $f'' + A(z)f = 0$ where $A(z)$ is entire, Ann. Acad. Sci. Fenn. Ser. A I Math. 11 (1986), no. 2, 275–294.
  • ––––, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988), no. 1, 88–104.
  • ––––, Finite order solutions of second order linear differential equations, Trans. Amer. Math. Soc. 305 (1988), no. 1, 415–429.
  • H. Habib and B. Belaïdi, Growth of solutions to higher-order linear differential equations with entire coefficients, Electron. J. Differential Equations 2014 (2014), no. 114, 17 pp.
  • W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs Clarendon Press, Oxford, 1964.
  • 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.
  • E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley Publ. Co., Reading, Mass.-London-Don Mills, Ont., 1969.
  • P. B. Kennedy, A class of integral functions bounded on certain curves, Proc. London Math. Soc. (3) 6 (1956), no. 4, 518–547.
  • I. Laine, Nevanlinna Theory and Complex Differential Equations, De Gruyter Studies in Mathematics 15, Walter de Gruyter, Berlin, 1993.
  • I. Laine and R. Yang, Finite order solutions of complex linear differential equations, Electron. J. Differential Equations 65 (2004), 1–8.
  • J. Long, C. Qiu and P. Wu, On the growth of solutions of a class of higher order linear differential equations with extremal coefficients, Abstr. Appl. Anal. 2014, Art. ID 305710, 7 pp.
  • J. Long, P. 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.
  • J. Long, J. Zhu and X. Li, Growth of solutions to some higher-order linear differential equations, Acta Math. Sci. Ser. A Chin. Ed. 33 (2013), no. 3, 401–408.
  • M. Ozawa, On a solution of $w'' + e^{-z}w' + (az+b)w = 0$, Kodai Math. J. 3 (1980), no. 2, 295–309.
  • K. C. Shin, New polynomials $P$ for which $f'' + P(z)f = 0$ has a solution with almost all real zeros, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 491–498.
  • M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.
  • H. Wittich, Zur Theorie linearer Differentialgleichungen im Komplexen, Ann. Acad. Sci. Fenn. Ser. A I Math. 379 (1966), 19 pp.
  • S. Wu, Some results on entire functions of finite lower order, Acta Math. Sinica (N.S.) 10 (1994), no. 2, 168–178.
  • X. Wu, J. Long, J. Heittokangas and K. Qiu, On second order complex linear differential equations with special functions or extremal functions as coefficients, Electronic J. Differential Equa. 2015 (2015), no. 143, 1–15.
  • L. Yang, Deficient values and angular distribution of entire functions, Trans. Amer. Math. Soc. 308 (1988), no. 2, 583–601.
  • ––––, Value Distribution Theory, Springer-Verlag, Berlin, 1993.
  • L. Yang and G. H. Zhang, Distribution of Borel directions of entire functions, Acta Math. Sinica 19 (1976), no. 3, 157–168.
  • G. H. Zhang, On entire functions extremal for Denjoy's conjecture, Sci. Sinica 24 (1981), no. 7, 885–898.
  • ––––, Theory of Entire and Meromorphic Functions-Deficient and Asymptotic Values and Singular Directions, Springer-Verlag, Berlin, 1993.