On the growth of subharmonic functions along paths
Let u be a subharmonic function on \({\mathbb{C}}\) such that \(\lim_{r\to\infty }M(r)/\log r=\infty,\) where \(M(r)=\max\{u(z):| z| =r\}.\) \textit{A. Huber} [Comment. Math. Helv. 32, 13-72 (1957; Zbl 0080.150)] proved that if \(\lambda >0\), then there exists a path \(\Gamma\) (\(\lambda)\) tending to \(\infty\) such that \(\int_{\Gamma (\lambda)}e^{-\lambda u} | dz| <\infty,\) and \textit{M. N. M. Talpur} [Proc. Lond. Math. Soc., III. Ser. 32, 193-198 (1976; Zbl 0329.31001)] proved that there is a path \(\Gamma\) tending to \(\infty\) such that \(u(z)/\log| z|\to \infty\) as \(z\to\infty \) on \(\Gamma\). The authors generalize these theorems by showing that there exists a path tending to \(\infty\) such that \(\int_{\Gamma}e^{-\lambda u} | dz| <\infty\) for each \(\lambda >0\) and \(u(z)/\log| z|\to \infty\) as \(z\to\infty \) on \(\Gamma\). The important feature of this result is that \(\Gamma\) is independent of \(\lambda\). The main tool is a new version of Hall's lemma. The authors give an indication of how their methods can be used to obtain the following form of a theorem of Davis and Lewis (to appear): Let D be a simply connected domain in C with 0\(\in D\). If v is subharmonic in D with 0\(\leq v\leq 1\) in D and \(v(0)=\epsilon >0\), then there is a path \(\gamma\) from 0 to some point \(b\in\partial D\) with \(\gamma \backslash\{b\}\subset D\) such that \(v>0\) on \(\gamma \backslash\{b\}\) and \(length(\gamma)\leq\alpha \epsilon^{-\beta} dist(0,\partial D),\) where \(\alpha\) and \(\beta\) are positive absolute constants.
scientific article; zbMATH DE number 3871752