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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 c0, 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 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:
30D35Distribution of values (one complex variable); Nevanlinna theory