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On the growth of meromorphic functions of infinite order. (English) Zbl 0828.30013
For a meromorphic function \(f\) let \(T(r,f)\) denote its Nevanlinna (or Ahlfors-Shimizu) characteristic and let \(M(r,f)\) denote its maximum modulus. In this paper, it is proved that if \(\gamma\) is an increasing and differentiable function with \(T(r,f)\leq \gamma(r)\) for large \(r\), then \[ \liminf_{r\to \infty} {{\log M(r,f)} \over {r\gamma' (r)}} \leq \pi \] and that if \(\psi\) is a positive and continuously differentiable function such that \(\psi(x) /x\) is nondecreasing, \(\psi' (x)\leq \sqrt {\psi (x)}\), and \(\int_{x_0}^\infty dx/ \psi(x)< \infty\), then \[ \liminf_{r\to \infty} {{\log M(r,f)} \over {T(r,f) \psi (\log T(r,f))}} =0. \] The proofs of these results are based on the method of Petrenko which was modified by W. H. J. Fuchs [Topics in Nevanlinna theory, Washington D.C., Proc. NRL Conf. Classical function theory, 1-32 (1970; Zbl 0294.30021)]. The proof in Fuchs’ paper uses Pólya-peaks which exist in general only for meromorphic functions of finite lower order. For the proofs of the results in this paper the Pólya-peaks are replaced are replaced by a suitable other sequence of \(r\)-values which can be considered as Pólya-peaks of infinite order.
C. J. Dai, D. Drasin and B. Q. Li [J. Anal. Math. 55, 217-228 (1990; Zbl 0722.30016); Correction: J. Anal. Math. 57, 299-300 (1991; Zbl 0767.30027)] have shown that \[ \lim_{r\to \infty} {{\log M(r,f)} \over {T(r,f) \varphi (\log T(r,f)) \log \varphi (\log T(r,f))}} =0 \] on a set of logarithmic density 1, where \(\varphi\) is an increasing, positive function with \(\int_{x_0}^\infty {{dx} \over {\varphi (x)}} <\infty\).
For a meromorphic function \(f\) and a complex number \(a\) let \(b(a,f):= \liminf_{r\to \infty} {{\log M(r, 1/(f- a))} \over {rT' (r,f)}}\), \(b(\infty, f):= \liminf_{r\to \infty} {{\log M(r,f)} \over {rT' (r,f)}}\). The first result of this paper shows that \(b(\infty, f)\leq \pi\) for functions of infinite order. Recently, A. Eremenko has proved that for every function \(f\) with lower order greater than \(1/2\) the set \(\{a\in \widehat {\mathbb{C}}\); \(b(a,f)> 0\}\) is countable and that \(\sum_{a\in \widehat {\mathbb{C}}} b(a,f)\leq 2\pi\).
Reviewer: G.Jank (Aachen)

30D30 Meromorphic functions of one complex variable, general theory
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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