×

Uniqueness of meromorphic functions sharing values with their shifts. (English) Zbl 1217.30029

Two meromorphic functions \(f\) and \(g\) are said to share a value or a function \(a\) if \(f-a\) and \(g-a\) have the same zeros. One distinguishes the cases whether the values (or functions) are shared counting multiplicities (CM) or ignoring multiplicities (IM). A famous theorem of Nevanlinna says that two meromorphic functions are equal if they share five values IM, and they differ only by a Möbius transformation if they share four values CM [R. Nevanlinna, Acta Math. 48, 367–391 (1926; JFM 52.0323.03)].
There is a vast literature on meromorphic functions sharing values with differential polynomials.
Here, the authors consider the case that a meromorphic function \(f(z)\) and the shift \(f(z+c)\), where \(c\neq 0\), share values or functions.
It is shown that if \(f\) is a meromorphic function of finite order, and if \(f(z)\) and \(f(z+c)\) share three values \(a_1,a_2,a_3\) CM, then \(f(z)=f(z+c)\). In fact, the values \(a_j\) may be replaced by periodic meromorphic functions satisfying \(T(r,a_j)=o(T(r,f))\). If \(\infty\) is a deficient value of \(f\), then two values or functions \(a_1,a_2\) suffice. In particular, this is the case for entire \(f\). The number of shared values may further be reduced if \(f\) has also a finite deficient value.
Finally it is shown that if \(f(z)\), \(f(z+c_1)\) and \(f(z+c_2)\) share three values CM, where \(c_1,c_2\) are linearly independent over the reals, then \(f\) is an elliptic function.

MSC:

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

Citations:

JFM 52.0323.03
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Yang C-C, Mathematics and its Applications 557 (2003)
[2] Nevanlinna R, Le Théorème de Picard-Borel et la Théorie des Fonctions Meromorphes (1929)
[3] DOI: 10.1112/jlms/s2-20.3.457 · Zbl 0413.30025
[4] DOI: 10.1080/17476939208814547 · Zbl 0776.30018
[5] Steinmetz N, Ann. Acad. Sci. Fenn. Ser. A I Math. 13 pp 93– (1988)
[6] DOI: 10.1090/S0002-9947-1983-0694375-0
[7] DOI: 10.2996/kmj/1138036659 · Zbl 0518.30029
[8] Gundersen GG, Pacific J. Math. 105 pp 299– (1983)
[9] Mues E, Resultate Math. 6 pp 48– (1983)
[10] DOI: 10.1016/0022-247X(80)90092-X · Zbl 0447.30018
[11] DOI: 10.1007/BF01303627 · Zbl 0416.30028
[12] Frank G, Michigan Math. J. 49 pp 175– (1999)
[13] Mues E, Results Math. 22 pp 725– (1992)
[14] Brosch G, Eindeutigkeitssätze für meromorphe funktionen (1989)
[15] DOI: 10.1007/BF01844071 · Zbl 0384.30009
[16] DOI: 10.1112/jlms/s2-4.2.357 · Zbl 0223.30036
[17] Halburd RG, Ann. Acad. Sci. Fenn. Math. 31 pp 463– (2006)
[18] DOI: 10.1007/s11139-007-9101-1 · Zbl 1152.30024
[19] DOI: 10.1112/plms/pdl012 · Zbl 1119.39014
[20] Clunie J, Mathematical Essays Dedicated to A. J. McIntyre pp 75– (1970)
[21] DOI: 10.1007/BF02392741 · Zbl 1203.30035
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.