Proceedings of the Japan Academy, Series A, Mathematical Sciences

Entire functions sharing an entire function of smaller order with their shifts

Xiao-Min Li, Xiao Yang, and Hong-Xun Yi

Full-text: Open access

Abstract

We study the growth of solutions of a certain difference equations, and study the uniqueness question of entire functions of finite orders sharing an entire function of smaller order with their shifts. The uniqueness results in this paper also extend and improve Theorem 1 [11].

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 89, Number 2 (2013), 34-39.

Dates
First available in Project Euclid: 30 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.pja/1359554919

Digital Object Identifier
doi:10.3792/pjaa.89.34

Mathematical Reviews number (MathSciNet)
MR3024273

Zentralblatt MATH identifier
1317.30041

Subjects
Primary: 30D35: Distribution of values, Nevanlinna theory
Secondary: 30D20: Entire functions, general theory 30D30: Meromorphic functions, general theory

Keywords
Difference Nevanlinna theory uniqueness of entire functions shared values

Citation

Li, Xiao-Min; Yang, Xiao; Yi, Hong-Xun. Entire functions sharing an entire function of smaller order with their shifts. Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), no. 2, 34--39. doi:10.3792/pjaa.89.34. https://projecteuclid.org/euclid.pja/1359554919


Export citation

References

  • A. H. H. Al-Khaladi, On meromorphic functions that share one value with their derivative, Analysis (Munich) 25 (2005), no. 2, 131–140.
  • R. B. Ash, Complex variables, Academic Press, New York, 1971.
  • R. Brück, On entire functions which share one value CM with their first derivative, Results Math. 30 (1996), no. 1–2, 21–24.
  • Y.-M. Chiang and S.-J. Feng, On the Nevanlinna characteristic of $f(z+\eta)$ and difference equations in the complex plane, Ramanujan J. 16 (2008), no. 1, 105–129.
  • Y.-M. Chiang and S.-J. Feng, On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions, Trans. Am. Math. Soc. 361 (2009), no. 7, 3767–3791.
  • 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.
  • R. G. Halburd and R. J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 2, 463–478.
  • R. G. Halburd and R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl. 314 (2006), no. 2, 477–487.
  • W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.
  • J. Heittokangas, R. Korhonen, I. Laine and J. Rieppo, Uniqueness of meromorphic functions sharing values with their shifts, Complex Var. Elliptic Equ. 56 (2011), no. 1–4, 81–92.
  • J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and J.-L. Zhang, Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity, J. Math. Anal. Appl. 355 (2009), no. 1, 352–363.
  • I. Laine, Nevanlinna theory and complex differential equations, de Gruyter Studies in Mathematics, 15, de Gruyter, Berlin, 1993.
  • I. Laine and C.-C. Yang, Clunie theorems for difference and $q$-difference polynomials, J. Lond. Math. Soc. (2) 76 (2007), no. 3, 556–566.
  • I. Laine and C.-C. Yang, Value distribution of difference polynomials, Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), no. 8, 148–151.
  • X.-M. Li, Entire functions sharing a finite set with their difference operators, Computational Methods and Function Theory 12 (2012), 307–328.
  • K. Liu, Meromorphic functions sharing a set with applications to difference equations, J. Math. Anal. Appl. 359 (2009), no. 1, 384–393.
  • K. Liu and L.-Z. Yang, Value distribution of the difference operator, Arch. Math. (Basel) 92 (2009), no. 3, 270–278.
  • A. I. Markushevich, Theory of functions of a complex variable. Vol. II, Revised English edition translated and edited by Richard A. Silverman, Prentice Hall, Englewood Cliffs, NJ, 1965.
  • J.-Y. Qiao, The value distribution of entire functions of finite order, Kodai Math. J. 12 (1989), no. 3, 429–436.
  • L. A. Rubel and C.-C. Yang, Values shared by an entire function and its derivative, in Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), 101–103. Lecture Notes in Math., 599, Springer, Berlin, 1977.
  • C.-C. Yang and H.-X. Yi, Uniqueness theory of meromorphic functions, Mathematics and its Applications, 557, Kluwer Acad. Publ., Dordrecht, 2003.
  • J.-L. Zhang, Value distribution and shared sets of differences of meromorphic functions, J. Math. Anal. Appl. 367 (2010), no. 2, 401–408.