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Zbl 1217.30029
Heittokangas, J.; Korhonen, R.; Laine, I.; Rieppo, J.
Uniqueness of meromorphic functions sharing values with their shifts. (English)
[J] Complex Var. Elliptic Equ. 56, No. 1-4, 81-92 (2011). ISSN 1747-6933; ISSN 1747-6941/e

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 [{\it R. Nevanlinna}, Acta Math. 48, 367--391 (1926; JFM 52.0323.03)]. \par There is a vast literature on meromorphic functions sharing values with differential polynomials. \par 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. \par 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. \par 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.
[Walter Bergweiler (Kiel)]

MSC 2000:: *30D35 Distribution of values (one complex variable)

Keywords: meromorphic function; sharing values

Citations: JFM 52.0323.03

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