Analytic functions with large sets of Fatou points

Abstract:

For a function $f$ analytic in the unit disc $D$, and for each $\lambda > 0$, let $L\left ( \lambda \right ) = \left \{ {z \in D:\left | {f\left ( z \right )} \right | = \lambda } \right \}$ denote a level set for $f$. We introduce a class $\mathcal {L}$, of functions characterized by geometric properties of a collection of sets $\left \{ {L\left ( {{\lambda _n}} \right )} \right \}$, where $\left \{ {{\lambda _n}} \right \}$ is an unbounded sequence. We show that ${\mathcal {L}_1}$, is a proper subclass of the class $\mathcal {L}$ of G. R. MacLane. Let ${A_\infty }$ denote the set of points ${e^{i\theta }}$ at which the function $f$ has $\infty$ as an asymptotic value, and let $F\left ( f \right )$ denote the set of Fatou points of $f$. We prove that for a function $f$ in the class ${\mathcal {L}_1}$, if $\Gamma$ is an arc of the unit circle such that $\Gamma \cap {A_\infty } = \emptyset$, then almost every point of $\Gamma$ belongs to $F\left ( f \right )$.