an:00718525
Zbl 0828.30013
Bergweiler, Walter; Bock, Heinrich
On the growth of meromorphic functions of infinite order
EN
J. Anal. Math. 64, 327-336 (1994).
00023194
1994
j
30D30 30D35
growth properties; maximum modulus; Nevanlinna characteristic
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 \textit{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 P??lya-peaks which exist in general only for meromorphic functions of finite lower order. For the proofs of the results in this paper the P??lya-peaks are replaced are replaced by a suitable other sequence of \(r\)-values which can be considered as P??lya-peaks of infinite order.
\textit{C. J. Dai}, \textit{D. Drasin} and \textit{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 \(\sum_{a\in \widehat {\mathbb{C}}} b(a,f)\leq 2\pi\).
G.Jank (Aachen)
Zbl 0294.30021; Zbl 0722.30016; Zbl 0767.30027