On sharing values of meromorphic functions and their differences. (English) Zbl 1267.30077

The authors obtain a difference analogue of the Brück conjecture and some results on uniqueness. They prove that \(a=0\) and \[ \frac{f(z+\eta)-f(z)}{f(z)}=A, \] where \(A\) is a nonzero constant, if \(f(z)\) is a finite order transcendental entire function which has a finite Borel exceptional value \(a\) and \(\Delta f(z)=f(z+\eta)-f(z)\not\equiv 0\), and if \(\Delta f\) and \(f\) share the value \(a\) CM. Some other results are also obtained. These contributions are interesting and new, and make the Brück conjecture more popular.


30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
39A10 Additive difference equations
Full Text: DOI


[1] Brück R.: On entire functions which share one value CM with their first derivative. Results Math. 30, 21–24 (1996) · Zbl 0861.30032
[2] 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) · Zbl 1062.30032
[3] Chiang Y.M., Feng S.J.: On the Nevanlinna characteristic of f(z + {\(\eta\)}) and difference equations in the complex plane. Ramanujan J. 16, 105–129 (2008) · Zbl 1152.30024
[4] Gross F.: Factorization of Meromorhpic Functions. U.S. Government Printing Office, Washinton (1972) · Zbl 0266.30006
[5] Gundersen G.: Meromorphic functions that share four values. Trans. Am. Math. Soc. 277, 545–567 (1983) · Zbl 0508.30029
[6] Gundersen G.: Correction to Meromorphic functions that share four values Trans. Am. Math. Soc. 304, 847–850 (1987) · Zbl 0631.30028
[7] 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) · Zbl 0911.30022
[8] Hayman W.K.: Meromorphic Functions. Clarendon Press, Oxford (1964) · Zbl 0115.06203
[9] 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) · Zbl 1180.30039
[10] Heittokangas, J., Korhonen, R., Laine, I., Rieppo, J.: Uniqueness of meromorphic functions sharing values with their shifts. Complex Var. Elliptic Equ. (2011, in press) · Zbl 1217.30029
[11] Liu K.: Meromorphic functions sharing a set with applications to difference quations. J. Math. Anal. Appl. 359, 384–393 (2009) · Zbl 1177.30035
[12] Mues E.: Meromorphic functions sharing four valus. Complex Var. Elliptic Equ. 12, 167–179 (1989) · Zbl 0699.30025
[13] Nevanlinna R.: Einige Eindentigkeitssätze in der theorie der meromorphen funktionen. Acta Math. 48, 367–391 (1926) · JFM 52.0323.03
[14] Ozawa M.: On the existence of prime periodic entire functions. Kodai Math. Sem. Rep. 29, 308–321 (1978) · Zbl 0402.30025
[15] Yang L.: Value Distribution Theory. Science Press, Beijing (1993) · Zbl 0790.30018
[16] Yang C.C., Yi H.X.: Uniqueness Theory of Meromorphic Functions. Kluwer Academic Publishers Group, Dordrecht (2003) · Zbl 1070.30011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.