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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)

##### MSC:
 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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##### References:
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