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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 \(\varphi f^{(k)}\not\equiv 0\).

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