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

Jank, Gerhard; Mues, Erwin; Volkmann, Lutz

Complex Variables, Theory Appl. 6, 51-71 (1986).

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