Communications in Mathematical Analysis

Unicity of Entire Functions and a Related Problem

Qi Han and Jingbo Liu

Full-text: Open access

Abstract

This paper concerns the unicity of entire functions and the growth estimate of entire solutions to certain complex linear non-homogenous differential equations.

Article information

Source
Commun. Math. Anal., Volume 9, Number 2 (2010), 42-50.

Dates
First available in Project Euclid: 3 June 2010

Permanent link to this document
https://projecteuclid.org/euclid.cma/1275586731

Mathematical Reviews number (MathSciNet)
MR2737753

Zentralblatt MATH identifier
1195.30040

Subjects
Primary: 30D20: Entire functions, general theory
Secondary: 34M05 30D35: Distribution of values, Nevanlinna theory

Keywords
entire function complex linear differential equation Nevanlinna theory order (& hyper-order) sharing value small function Wiman-Valiron estimate

Citation

Han, Qi; Liu, Jingbo. Unicity of Entire Functions and a Related Problem. Commun. Math. Anal. 9 (2010), no. 2, 42--50. https://projecteuclid.org/euclid.cma/1275586731


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References

  • C. Ai, A correction to a result of Frank-Hua on the unicity of meromorphic functions. Master Thesis of Shandong University, 2006, Jinan, Shandong, P.R. China.
  • C.A. Bernstein, D.C. Chang and B.Q. Li, Shared values for meromorphic functions. Adv. Math. 115 (1995), pp 201-220.
  • C.A. Bernstein, D.C. Chang and B.Q. Li, On uniqueness of entire functions in $\mathbb{C}^n$ and their partial differential polynomials. Forum Math. 8 (1996), pp 379-396.
  • C.A. Bernstein, D.C. Chang and B.Q. Li, The uniqueness problem and meromorphic solutions of partial differential equations. J. Anal. Math. 77 (1999), pp 51-68.
  • J.M. Chang and M.L. Fang, Uniqueness of entire functions. J. Math. Anal. Appl. 288 (2003), pp 97-111.
  • J.M. Chang and M.L. Fang, Entire functions that share a small function with their derivatives. Complex Var. Theory Appl. 49 (2004), pp 871-895.
  • G.G. Gundersen, Estamites for the logarithmic derivative of a meromorphic function, plus similar estimates. J. London Math. Soc. 37 (1988), pp 88-104.
  • G.G. Gundersen and E. Steinbart, Finite order solutions of non-homogenous linear differential equations. Ann. Acad. Sci. Fenn. Math. 17 (1992), pp 327-341.
  • G.G. Gundersen, E. Steinbart and S.P. Wang, The possible orders of solutions of linear differential equations with polynomial coefficients. Trans. Amer. Math. Soc. 350 (1998), pp 1225-1247.
  • G.G. Gundersen, E. Steinbart and S.P. Wang, Solutions of non-homogenous linear differential equations with exceptionally zeros. Ann. Acad. Sci. Fenn. Math. 23 (1998), pp 429-452.
  • G.G. Gundersen, E. Steinbart and S.P. Wang, Growth and oscillation of non-homogenous linear differential equations. Proc. Edinb. Math. Soc. 43 (2000), pp 343-359.
  • Q. Han and H.X. Yi, On the uniqueness problems of entire functions and their linear differential polynomials. Kodai Math. J. 30 (2007), pp 61-73.
  • W.K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
  • S. Hellerstein, J. Miles and J. Rossi, On the growth of solutions of certain linear differential equations. Ann. Acad. Sci. Fenn. Math. 17 (1992), pp 343-365.
  • G. Jank, E. Mues and L. Volkmann, Meromorphe funktionen, die mit iher ersten und zweiten Ableitung einen endlichen Wert teilen. Complex Var. Theory Appl. 6 (1986), pp 51-71.
  • I. Laine, Nevanlinna Theory and Complex Differential Equation, Walter de Gruyter, Berlin, New York, 1993.
  • P. Li and C.C. Yang, Uniqueness theorems on entire functions and their derivatives. J. Math. Anal. Appl. 253 (2001), pp 50-57.
  • P. Li and C.C. Yang, Further results on meromorphic functions that share two values with their defivatives. J. Aust. Math. Soc. 78 (2005), pp 91-102.
  • P. Li and W.J. Wang, Entire functions that share a small function with its derivative. J. Math. Anal. Appl. 328 (2007), pp 743-751.
  • L.A. Rubel and C.C. Yang, Values shared by an entire function and its derivative. Lecture Notes in Math. 599, pp 101-103, Springer-Verlag, New York, 1977.
  • S.P. Wang, On the frequency of zeros of a fundamental solution set of complex linear differential equations. Kodai Math. J. 20 (1997), pp 143-155.