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Slowly growing meromorphic functions and the zeros of differences. (English) Zbl 1201.30033

Considering a transcendental meromorphic function \(f\) in the plane such that \[ \liminf_{r\rightarrow\infty}T(r,f)/(\log r)^{2}=0, \] the authors prove that at least one of the functions \(F(z):=f(qz)-f(z)\) and \(G(z):=F(z)/f(z)\) has infinitely many zeros, provided the complex number \(q\) satisfies \(|q|>1\). The authors conjecture that in fact both of the functions \(F\) and \(G\) have infinitely many zeros. On the other hand, it is shown that the hypothesis concerning the growth of \(f\) is sharp. This follows by considering the function \(f(z)=\prod_{n=0}^{\infty}(1-z/q^{n})^{-1}\). A lemma of some independent interest, used to prove the main result, says that whenever \(f\) has either finitely many zeros or finitely many poles in the complex plane, then in fact \(G\) must have infinitely many zeros.

MSC:

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