an:01228255
Zbl 0942.30019
Marchenko, I. I.
Growth of entire and meromorphic functions
EN
Sb. Math. 189, No. 6, 875-899 (1998); translation from Mat. Sb. 189, No. 6, 59-84 (1998).
00051189
1998
j
30D35 30D30 30D20
meromorphic; Nevanlinna theory; deficiency; deviation; growth; order; spread; Polya peakes
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 \textit{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. \textit{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 \textit{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.
W.Bergweiler (Kiel)
Zbl 0828.30013; Zbl 0905.30025; Zbl 0716.30024; Zbl 0854.30023