Taiwanese Journal of Mathematics


Huifang Liu and Zhiqiang Mao

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In this paper, the uniqueness problems of entire functions and their difference operators are investigated. It is shown that if a finite order entire function $f$ shares $0,\alpha$ CM with its difference operator $\Delta_\eta f(z)=f(z+\eta)-f(z)$, then $\Delta_\eta f\equiv f$, where $\alpha$ is an entire function with order less than $f$. The research results also include a difference analogue of Brück conjecture, and extend some results in Chen-Yi Results Math., 63 (2013), 557-565).

Article information

Taiwanese J. Math., Volume 19, Number 3 (2015), 907-917.

First available in Project Euclid: 4 July 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30D35: Distribution of values, Nevanlinna theory 39A10: Difference equations, additive

entire function difference operator sharing value


Liu, Huifang; Mao, Zhiqiang. ON THE UNIQUENESS PROBLEMS OF ENTIRE FUNCTIONS AND THEIR DIFFERENCE OPERATORS. Taiwanese J. Math. 19 (2015), no. 3, 907--917. doi:10.11650/tjm.19.2015.4674. https://projecteuclid.org/euclid.twjm/1499133668

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  • R. Brück, On entire functions which share one value CM with their first derivative, Results Math., 30 (1996), 21-24.
  • Z. X. Chen and H. X. Yi, On sharing values of meromorphic functions and their differences, Results Math., 63 (2013), 557-565.
  • 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), 105-129.
  • Y. M. Chiang and S. J. Feng, On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions, Trans. Amer. Math. Soc., 361 (2009), 3767-3791.
  • G. Gundersen and L. Z. Yang, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl., 223 (1998), 88-95.
  • G. G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc., 37 (1988), 88-104.
  • W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
  • J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and J. Zhang, Value sharing results for shifts of meromorphic functions, and suffficient conditions for periodicity, J. Math. Anal. Appl., 355 (2009), 352-363.
  • J. Heittokangas, R. Korhonen, I. Laine and J. Rieppo, Uniqueness of meromorphic functions sharing values with their shifts, Complex Var. Elliptic Equ., 56 (2011), 81-92.
  • K. Ishizaki and K. Tohge, On the complex oscillation of some linear differential equations, J. Math. Anal. Appl., 206 (1997), 503-517.
  • I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993.
  • K. Liu and L. Z. Yang, Value distribution of the difference operator, Arch. Math., 92 (2009), 270-278.
  • X. M. Li and H. X. Yi, Entire functions sharing an entire function of smaller order with their difference operators, Acta Math. Sin. English Series, 30 (2014), 481-498.
  • E. Mues and N. Stinmetz, Meromorphe funktionen, die mit ihrer ableitung werte teilen, Manuscripta Math., 29 (1979), 195-206.
  • L. Z. Yang, Solution of a differential equation and its applications, Kodai Math. J., 22 (1999), 458-464.
  • C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, New York, 2003.