The growth of entire and harmonic functions along asymptotic paths (Q1061264)

The authors continue work of \textit{J. Lewis} and the authors, Ark. Mat. 22, 109-119 (1984; Zbl 0547.31003) and the first author, J. Lond. Math. Soc., II. Ser. 30, 73-78 (1984; Zbl 0554.31001). Theorem 1. let f be an entire function such that for some \(K>0\), at least one of the level curves \(| f| =K\) tends to \(\infty\). Then there exists a path \(\Gamma\) from 0 to \(\infty\) such that \[ (*)\quad \log | f(z)| >| z|^{-o(1)} \] (z\(\to \infty\); \(z\in \Gamma)\), and \[ (**)\quad \int_{\Gamma}(\log | f|)^{-2-\lambda}| dz| <\infty \quad for\quad all\quad \lambda >0. \] Corollary 1. If u is a nonconstant harmonic function in \({\mathbb{C}}\) then there exists a curve \(\Gamma\) form 0 to \(\infty\) such that (*) and (**) hold with log \(| f|\) replaced by u. Theorem 2. Let f be an entire function of order \(\rho\leq \infty\) such that for some \(K>0\) the set \(\{| f| >K\}\) contains at least two components. Then there exists a path \(\Gamma\) from 0 to \(\infty\) such that log \(| f(z)| >| z|^{-(\rho /(2\rho -1))-o(1)}\) as \(z\to \infty\), \(z\in \Gamma\) and \[ \int_{\Gamma}(\log | f|)^{-((2\rho -1)/\rho)+\lambda}| dz| <\infty \quad for\quad all\quad \lambda >0. \] For some refinements of these results and further work of this type see \textit{J. M. Wu}, Length of paths for subharmonic functions (to appear).