Abstract
In this paper, we firstly consider the Brück conjecture itself and show that it holds exactly for the entire function \({f(z)=\frac{1}{c}(Ae^{cz}-a)+a}\) , where A, a, c are nonzero constants. Then we give a necessary and sufficient condition that f(z) and f ′(z) share a finite value a CM for some special cases. Finally, we investigate two analogues of the Brück conjecture including the difference analogue of the Brück conjecture raised by Liu and Yang (Arch. Math. 92, 270–278 (2009)) and the shifted analogue of the Brück conjecture raised by Heittokangas et al. (J. Math. Anal. Appl. 355, 352–363 (2009)). And we give some necessary conditions when f(z) shares a finite value a CM with its difference operators or shifts.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bergweiler W., Langley J.K.: Zeros of differences of meromorphic functions. Math. Proc. Camb. Phil. Soc. 142, 133–147 (2007)
Brück R.: On entire functions which share one value CM with their first derivative. Results Math. 30, 21–24 (1996)
Chen Z.X.: The growth of solutions of f′′ + e −z f′ + Q(z)f = 0 where the order of Q = 1. Science in China (Ser.A) 45, 290–300 (2002)
Chiang Y.M., Feng S.J.: On the Nevanlinna characteristic f(z + η) and difference equations in the complex plane. Ramanujan J. 16, 105–129 (2008)
Chiang Y.M., Feng S.J.: On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. Trans. Amer. Math. Soc. 361, 3767–3791 (2009)
Gundersen G.G.: Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J. London Math. Soc. 37, 88–104 (1988)
Gundersen G.G., Yang L.Z.: Entire functions that share one value with one or two of their derivatives. J. Math. Anal. Appl. 223, 88–95 (1998)
Halburd R.G., Korhonen R.J.: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 314, 477–487 (2006)
Hayman W.: Meromorphic Functions. Clarendon Press, Oxford (1964)
Heittokangas J. et al.: Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity. J. Math. Anal. Appl. 355, 352–363 (2009)
G. Jank and L. Volkmann, Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, Birkhäuser, Basel, Boston, 1985.
Laine I.: Nevanlinna Theory and Complex Differential Equations. Walter de Gruyter, Berlin (1993)
Liu K., Yang L.Z.: Value distribution of the difference operator. Arch. Math. 92, 270–278 (2009)
Markushevich A.I.: Theory of functions of a complex variable, Vol. II. Prentice-Hall, Englewood Cliffs, New Jersey (1965)
Ozawa M.: On the existence of prime periodic entire functions. Kodai Math. Sem. Rep. 29, 308–321 (1978)
L. A. Rubel and C. C. Yang, Values shared by an entire function and its derivative, Lecture Notes in Math. 599, Berlin, Springer-Verlag, 101–103 (1977).
Yang L.: Value Distribution Theory and New Research. Science Press, Beijing (1982) (in Chinese)
C. C. Yang and H. X. Yi, The Uniqueness Theory of Meromorphic Functions, Math. Appl., 557, Kluwer Academic Publishers Group, Dordrecht, 2003.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by National Natural Science Fund of China No. 10771011 and the Fundamental Research Funds for the Central Universities NO. 300414. The first author is also supported by the Innovation Foundation of BUAA for Ph.D. Candidates.
Rights and permissions
About this article
Cite this article
Li, S., Gao, Z. A note on the Brück conjecture. Arch. Math. 95, 257–268 (2010). https://doi.org/10.1007/s00013-010-0165-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-010-0165-6