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.


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


JFM 52.0323.03
Full Text: DOI


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