*(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 $f\left(z\right)$ and the shift $f(z+c)$, where $c\ne 0$, share values or functions.

It is shown that if $f$ is a meromorphic function of finite order, and if $f\left(z\right)$ and $f(z+c)$ share three values ${a}_{1},{a}_{2},{a}_{3}$ CM, then $f\left(z\right)=f(z+c)$. In fact, the values ${a}_{j}$ may be replaced by periodic meromorphic functions satisfying $T(r,{a}_{j})=o\left(T(r,f)\right)$. 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 $f\left(z\right)$, $f(z+{c}_{1})$ 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 | Distribution of values (one complex variable); Nevanlinna theory |