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A note on the product of meromorphic functions and its derivatives. (English) Zbl 1007.30024
Bergweiler gave a partial answer to a question asked by this reviewer about whether it is true that for any non-constant meromorphic function $$f$$ in the plane, the function $$ff'-c$$, where $$c$$ is an arbitrary meromorphic function that satisfies $$T(r,c)=o(T(r,f))$$, must have an infinite number of zeros [W. Bergweiler, Bull. Hong Kong Math. Soc. 1, 97-101 (1996; Zbl 0928.30016)]. Here the $$T(r,f)$$ is the standard Nevanlinna characteristic function of a meromorphic function. His proof dealt with a finite order meromorphic function $$f$$ and a polynomial $$c$$.
The author of this paper under review solved the problem by giving an estimate of the $$T(r, f)$$ which may be of independent interest: $\begin{split} T(r, f)<N\left(r,{1\over f}\right)+N\left(r,{1\over \varphi f^{(k)}-a}\right)+N\left(r,{1\over \varphi f^{(k)}-b}\right)-\\ N(r,f)-N\left(r,{1\over (\varphi f^{(k)})^\prime}\right)+S(r, f),\end{split}$ as $$r\to +\infty$$, where $$k$$ is a positive integer, $$a$$ and $$b$$ are distinct non-zero constants, and $$\varphi$$ is any meromorphic function with $$T(r,\varphi)=o(T(r, f))$$ and $$\varphi f^{(k)}\not\equiv 0$$.

##### MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
##### Keywords:
zeros of meromorphic functions; derivatives
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