## 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

### Keywords:

meromorphic function; sharing values

JFM 52.0323.03
Full Text:

### References:

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