×

Value distribution and shared sets of differences of meromorphic functions. (English) Zbl 1188.30044

Summary: We investigate value distribution and uniqueness problems for difference polynomials of meromorphic functions. In particular, we show that, for a finite order transcendental meromorphic function \(f\) with \(\lambda (1/f)<\rho (f)\) and a non-zero complex constant \(c\), if \(n\geqslant 2\) then \(f(z)^nf(z+c)\) assumes every non-zero value \(a\in \mathbb C\) infinitely often. This research also shows that there exist two sets \(S_{1}\) with \(9\) (resp. \(5\)) elements and \(S_{2}\) with \(1\) element such that, for a finite order non-constant meromorphic (resp. entire) function \(f\) and a non-zero complex constant \(c\), \(E_{f(z)}(S_j)=E_{f(z+c)}(S_j)\) \( (j=1,2)\) implies \(f(z)\equiv f(z+c)\). This gives an answer to a question of F. Gross [Complex Anal., Proc. Conf., Lexington 1976, Lect. Notes Math. 599, 51–67 (1977; Zbl 0357.30007)] concerning a finite order meromorphic function \(f\) and its shift.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

Citations:

Zbl 0357.30007
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bergweiler, W.; Langley, J. K., Zeros of difference of meromorphic functions, Math. Proc. Cambridge Philos. Soc., 142, 133-147 (2007) · Zbl 1114.30028
[2] 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
[3] Clunie, J., On a result of Hayman, J. London Math. Soc., 42, 389-392 (1967) · Zbl 0169.40801
[4] Fang, M. L., Uniqueness and value-sharing of entire functions, Comput. Math. Appl., 44, 823-831 (2002) · Zbl 1035.30017
[5] Frank, G.; Reinders, M., A unique range set for meromorphic functions with 11 elements, Complex Var. Theory Appl., 37, 185-193 (1998) · Zbl 0952.30029
[6] Gross, F., Factorization of meromorphic functions and some open problems, (Complex Analysis, Proc. Conf. Univ. Kentucky. Complex Analysis, Proc. Conf. Univ. Kentucky, Lexington, KY, 1976. Complex Analysis, Proc. Conf. Univ. Kentucky. Complex Analysis, Proc. Conf. Univ. Kentucky, Lexington, KY, 1976, Lecture Notes in Math., vol. 599 (1997), Springer: Springer Berlin), 51-69
[7] Gross, F.; Osgood, C. F., Entire functions with common preimages, (Factorization Theory of Meromorphic Functions (1982), Marcel Dekker), 19-24 · Zbl 0494.30029
[8] Halburd, R. G.; Korhonen, R. J., Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn., 31, 463-478 (2006) · Zbl 1108.30022
[9] 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) · Zbl 1085.30026
[11] Heittokangas, J.; Korhonen, R.; Laine, I.; Rieppo, J.; Zhang, J., Value sharing results for shifts of meromorphic function, and sufficient conditions for periodicity, J. Math. Anal. Appl., 355, 352-363 (2009) · Zbl 1180.30039
[12] Hayman, W. K., Picard values of meromorphic functions and their derivatives, Ann. Math., 70, 9-42 (1959) · Zbl 0088.28505
[13] Hayman, W. K., Meromorphic Functions (1964), Clarendon Press: Clarendon Press Oxford · Zbl 0115.06203
[14] Laine, I., Nevanlinna Theory and Complex Differential Equations (1993), Walter de Gruyter: Walter de Gruyter Berlin
[15] Laine, I.; Yang, C. C., Value distribution of difference polynomials, Proc. Japan Acad. Ser. A, 83, 148-151 (2007) · Zbl 1153.30030
[17] Li, P.; Yang, C. C., Some further results on the unique range sets of meromorphic functions, Kodai Math. J., 18, 437-450 (1995) · Zbl 0849.30025
[18] Lin, W. C.; Yi, H. X., Uniqueness theorems for meromorphic functions, Indian J. Pure Appl. Math., 52, 2, 121-132 (2004) · Zbl 1056.30031
[19] Lin, W. C.; Yi, H. X., Uniqueness theorems for meromorphic function concerning fixed-points, Complex Var. Theory Appl., 49, 11, 793-806 (2004) · Zbl 1067.30065
[20] Liu, K.; Yang, L. Z., Value distribution of the difference operator, Arch. Math., 92, 270-278 (2009) · Zbl 1173.30018
[21] Mues, E.; Reinders, M., Meromorphic functions sharing one value and unique range sets, Kodai Math. J., 18, 515-522 (1995) · Zbl 0919.30023
[22] Yang, C. C.; Hua, X. H., Uniqueness and value-sharing of meromorphic functions, Ann. Acad. Sci. Fenn. Math., 22, 395-406 (1997) · Zbl 0890.30019
[23] Yang, C. C.; Yi, H. X., Uniqueness Theory of Meromorphic Functions (1995), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht: Science Press: Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht: Science Press Beijing, Chinese original: · Zbl 0799.30019
[24] Yi, H. X., On a question of Gross, Sci. China Ser. A, 38, 8-16 (1995) · Zbl 0819.30017
[25] Yi, H. X.; Yang, L. Z., Meromorphic functions that share two sets, Kodai Math. J., 20, 127-134 (1997) · Zbl 0891.30019
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.