Abstract
For a meromorphic function f in the complex plane, we prove that if f is a finite order transcendental entire function which has a finite Borel exceptional value a, if \({f(z+\eta)\not\equiv f(z)}\) for some \({\eta\in \mathbb{C}}\), and if f(z + η) − f(z) and f(z) share the value a CM, then
where A is a nonzero constant. We also consider problems on sharing values of meromorphic functions and their differences when their orders are not an integer or infinite.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Brück R.: On entire functions which share one value CM with their first derivative. Results Math. 30, 21–24 (1996)
Chen Z.X., Shon K.H.: On conjecture of R. Brück, concerning the entire function sharing one value CM with its derivative Taiwanese. J. Math. 8(2), 235–244 (2004)
Chiang Y.M., Feng S.J.: On the Nevanlinna characteristic of f(z + η) and difference equations in the complex plane. Ramanujan J. 16, 105–129 (2008)
Gross F.: Factorization of Meromorhpic Functions. U.S. Government Printing Office, Washinton (1972)
Gundersen G.: Meromorphic functions that share four values. Trans. Am. Math. Soc. 277, 545–567 (1983)
Gundersen G.: Correction to Meromorphic functions that share four values Trans. Am. Math. Soc. 304, 847–850 (1987)
Gundersen G., Yang L.Z.: Entire functions that share one values with one or two of their derivatives. J. Math. Anal. Appl. 223(1), 88–95 (1998)
Hayman W.K.: Meromorphic Functions. Clarendon Press, Oxford (1964)
Heittokangas J., Korhonen R., Laine I., Rieppo J., Zhang J.: Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity. J Math. Anal. Appl. 355, 352–363 (2009)
Heittokangas, J., Korhonen, R., Laine, I., Rieppo, J.: Uniqueness of meromorphic functions sharing values with their shifts. Complex Var. Elliptic Equ. (2011, in press)
Liu K.: Meromorphic functions sharing a set with applications to difference quations. J. Math. Anal. Appl. 359, 384–393 (2009)
Mues E.: Meromorphic functions sharing four valus. Complex Var. Elliptic Equ. 12, 167–179 (1989)
Nevanlinna R.: Einige Eindentigkeitssätze in der theorie der meromorphen funktionen. Acta Math. 48, 367–391 (1926)
Ozawa M.: On the existence of prime periodic entire functions. Kodai Math. Sem. Rep. 29, 308–321 (1978)
Yang L.: Value Distribution Theory. Science Press, Beijing (1993)
Yang C.C., Yi H.X.: Uniqueness Theory of Meromorphic Functions. Kluwer Academic Publishers Group, Dordrecht (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the National Natural Science Foundation of China (No: 11171119, 11171184).
Rights and permissions
About this article
Cite this article
Chen, ZX., Yi, HX. On Sharing Values of Meromorphic Functions and Their Differences. Results. Math. 63, 557–565 (2013). https://doi.org/10.1007/s00025-011-0217-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-011-0217-7