On slowly increasing unbounded harmonic functions (Q1192242)

The author extends work of \textit{J. A. Jenkins} and \textit{K. Oikawa} [Acta Math. 124, 37-63 (1970; Zbl 0194.136)], itself based on the theory of multivalent functions due to W. K. Hayman and Hayman's students. In the Jenkins-Oikawa theory, \(u\) is harmonic in a plane domain \(\Omega\), with the growth of \(u\) measured by that of \(\mu(a,r)\equiv\int^ r_ a dt/\Theta(t)\), with \(\Theta(c)=\int_{\ell(c)}| * du|\), \(\ell(c)=\{z\in\Omega;\;u(z)=c\}\). In particular, Jenkins and Oikawa consider harmonic functions \(u\) of maximum growth in \(\Omega=\{| z|<1\}\): there is a direction \(e^{i\theta_ 0}\) and sequences \(r_ n\to 1\), \(b_ n\leq u(r_ n e^{i\theta_ 0})\) and, for some \(a\), \[ \limsup_{n\to\infty}\left\{\mu(a,b_ n)-{1\over \pi}\log\left[{1\over 1-r_ n}\right]\right\}>-\infty.\tag{*} \] In this work the author generalizes this situation to that where \(u\) has \((\alpha_ 1,\dots,\alpha_ k)\)-growth \((\sum\alpha_ j=1)\) at \(e^{i\theta_ 1},\dots,e^{i\theta_ k}\); corresponding to \((*)\) is that \[ \limsup_{n\to\infty}\left[\mu(a,b_ n)-{1\over\pi}\sum^ k_{n=i}\alpha^ 2_ j\log\left\{{1\over 1-r_ n[j]}\right\}\right]>- \infty, \leqno(**) \] where the \(r^{[j]}_ n\) \((j=1,\dots,k)\) tend to 1 appropriately. Jenkins-Oikawa had considered the case \(\alpha_ j\equiv 1/k\) \((j=1,\dots,k)\). As in work cited above, the author shows that \((**)\) implies that \(u\) grows rapidly and regularly near each ray \(\{\arg z=\theta_ j\}\) as \(| z|\to 1\), and applies his theory both to harmonic functions defined in strips, and to circumferentially mean 1-valent analytic functions \(f(z)\) in a ``\(k\)-star domain''; in this situation, \(\Omega\) may be divided into sectors in each of which \(f\) is unbounded, but no hypothesis is made about the growth of \(f\) along any specific collection of rays. The paper closes with an example of Warschawski type, to show that even in the simple case of one ray, such refined conclusions fail when \((*)\) is weakened to \(\mu(a,b_ n)={1\over\pi}(1+o(1))\log(1-r_ n)\) \((n\to\infty)\).