On the generalized Seidel class $U$

Abstract:

As usual, we say that a function $f \in U$ if $f$ is meromorphic in $| z | < 1$ and has radial limits of modulus $1$ a.e. (almost everywhere) on an arc $A$ of $\left | z \right | = 1$. This paper contains three main results: First, we extend our solution of A. J. Lohwater’s problem (1953) by showing that if $f \in U$ and $f$ has a singular point $P$ on $A$, and if $\upsilon$ and $1/\bar {\upsilon }$ are a pair of values which are not in the range of $f$ at $P$, then one of them is an asymptotic value of $f$ at some point of $A$ near $P$. Second, we extend our solution of J. L. Doob’s problem (1935) from analytic functions to meromorphic functions, namely, if $f \in U$ and $f(0) = 0$, then the range of $f$ over $\left | z \right | < 1$ covers the interior of some circle of a precise radius depending only on the length of $A$. Finally, we introduce another class of functions. Each function in this class has radial limits lying on a finite number of rays a.e. on $\left | z \right | = 1$, and preserves a sector between domain and range. We study the boundary behaviour and the representation of functions in this class.