THE SHARPNESS OF A CRITERION OF MACLANE FOR THE CLASS ${\cal A}$

Abstract

A holomorphic function defined in the unit disk $\Delta = \{z: |z|<1\}$ belongs to the MacLane class ${\cal A}$ if each point $\zeta$ of a dense subset of $\p \Delta$ is the endpoint of a curve $\gamma_{\zeta}$ (with

$\gamma_{\zeta}\setminus \zeta \subset \Delta)$ such that $f(z)$ tends to a limit (perhaps $\infty$) as $z \to \zeta$ on $\gamma_\zeta$. The classical Fatou theorem ensures that $f \in {\cal A}$ when $f$ is bounded. G. R. MacLane introduced ${\cal A}$ in [5], where he proved that $f \in {\cal A}$ if there is a set $E$ dense in $\p \Delta$ with \begin{equation}\int^1_0 (1-r) \log^+|f(re^{i\theta})| \, dr<\infty\qquad(\theta \in E).\end{equation}

For example, if $f$ is the modular function and $M(r)=\max_{|z|=r}|f(z)|$ its maximum modulus, then

$$\log M(r)\leq \log\frac{1}{1-r} +O(1),$$

so that (1.1) applies. An ample discussion of ${\cal A}$ is in [4, Chapter 10].