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Growth of entire and meromorphic functions. (English. Russian original) Zbl 0942.30019
Sb. Math. 189, No. 6, 875-899 (1998); translation from Mat. Sb. 189, No. 6, 59-84 (1998).
For a function $$f$$ meromorphic in the plane let $$b(\infty,f)=\liminf_{r\rightarrow \infty} \log M(r,f)/A(r,f)$$, where $$A(r,f)=rT'(r,f)$$. Here $$T(r,f)$$ is the Ahlfors-Shimizu characteristic. It was shown by H. Bock and the reviewer [J. Anal. Math. 64, 327-336 (1994; Zbl 0828.30013)] that if the order $$\lambda$$ of $$f$$ satisfies $${1\over{2}}\leq\lambda\leq\infty$$, then $$b(\infty,f)\leq \pi$$. The author improves this result for functions attaining the maximum modulus at more than one point. Let $$p(r,\infty,f)$$ be the number of components of the set $$\{\varphi:|f(re^{i\varphi})|>1\}$$ that contain a point $$\phi$$ such that $$M(r,f)=|f(re^{i\phi})|$$. Let $$p(\infty,f)=\liminf_{r\rightarrow\infty} p(r,\infty,f)$$. It is shown (Theorem 1) that if $${1\over{2}}\leq\lambda/p(\infty,f)\leq\infty$$, then $$b(\infty,f)\leq \pi/p(\infty,f)$$. The sharp upper bound for $$b(\infty,f)$$ is also given for the case $$\lambda/p(\infty,f)<{1\over{2}}$$. For $$a\in{\mathbb C}$$ let $$b(a,f)=b(\infty,1/(f-a))$$ and let $$\Delta:=\Delta(a,f)$$ be the Valiron deficiency of $$a$$. It is shown (Theorem 2) that $$b(a,f)\leq \pi\sqrt{\Delta(2-\Delta)}$$ if $${1\over{2}}\leq\lambda\leq\infty$$ or if $$0<\lambda<{1\over{2}}$$ and $$\sin \pi \lambda/2\geq\sqrt{\Delta/2}$$. The upper bound for $$b(a,f)$$ for the remaining case is also given. A. Eremenko [Complex Variables, Theory Appl. 34, No. 1-2, 83-97 (1997; Zbl 0905.30025)] has shown that if $${1\over{2}}\leq\lambda\leq\infty$$, then $$\sum_{a\in\overline{\mathbb C}}b(a,f)\leq 2\pi$$. Here it is shown for $$\Delta:=\Delta(0,f')$$ that $$\sum_{a\in\overline{\mathbb C}}b(a,f)\leq 2\pi\sqrt{\Delta(2-\Delta})$$ for meromorphic $$f$$ (Theorem 3) and $$\sum_{a\in{\mathbb C}}b(a,f)\leq \pi\sqrt{\Delta(2-\Delta)}$$ for entire $$f$$ (Theorem 4). The author points out that his method of proof is different from that of Eremenko. Finally the author proves two results relating $$b(\infty,f)$$ and $$p(\infty,f)$$ to the spread $\sigma(\infty,f)=\limsup_{r\rightarrow \infty}\operatorname {meas} \{\varphi:|f(re^{i\varphi})|>1\}.$ Results similar to those obtained in this paper, with $$b(\infty,f)$$ replaced by the Petrenko deviation $$\beta(\infty,f)=\liminf_{\rightarrow\infty}\log M(r,f)/T(r,f)$$, were obtained by the author and A. I. Shcherba in previous papers [see Mat. Sb. 181, No. 1, 3-24 (1990; Zbl 0716.30024) and Mat. Sb. 186, No. 3, 85-102 (1995; Zbl 0854.30023)]. The methods developed there, which in part are based on Baernstein’s star function, are extended and modified in order to obtain the results of the present paper.

##### MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory 30D30 Meromorphic functions of one complex variable, general theory 30D20 Entire functions of one complex variable, general theory
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