# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $f\left(z\right)$ and the shift $f\left(z+c\right)$, 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 $f\left(z+c\right)$ share three values ${a}_{1},{a}_{2},{a}_{3}$ CM, then $f\left(z\right)=f\left(z+c\right)$. In fact, the values ${a}_{j}$ may be replaced by periodic meromorphic functions satisfying $T\left(r,{a}_{j}\right)=o\left(T\left(r,f\right)\right)$. 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 $f\left(z\right)$, $f\left(z+{c}_{1}\right)$ and $f\left(z+{c}_{2}\right)$ 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
##### Keywords:
meromorphic function; sharing values