## Meromorphe Funktionen, die mit ihrer ersten und zweiten Ableitung einen endlichen Wert teilen. (Meromorphic functions which share some finite value with its first and second derivative).(German)Zbl 0603.30037

Two meromorphic functions f and g share the value a by counting multiplicities (CM) if the zeros of f-a and g-a (1/f and 1/g if $$a=\infty)$$ are the same with same multiplicities.
In the paper under review the following is proved: Theorem 1. Let g be an entire function which shares some finite nonzero value a CM with its derivative g’. If, moreover, $$g''(z_ 0)=a$$ whenever $$g(z_ 0)=a$$, then $$g'=g.$$
Theorem 2. Let f be meromorphic in the plane and assume that f, f’ and f” share the finite value $$a\neq 0$$ CM. Then $$f=f'$$. The example $$e^{2z}$$ shows that the assumption $$a\neq 0$$ may not be dropped. It is, however, open whether Theorem 1 holds also for meromorphic functions.
The proof of Theorem 2 is quite extensive. It requires to construct highly artistic auxiliary functions, such as $V=\frac{f\prime''(f- f')}{(f''-a)(f'-a)}+\frac{f\prime''}{f''-a}+\frac{f''}{f'-a}-\frac{f'- f''}{f-f'}$ and leads finally to the surprising contradiction $$4=35/9$$.
Reviewer: N.Seinmetz

### MSC:

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

### Keywords:

value-sharing; meromorphic functions
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