# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## On the generalized Seidel class $U$HTML articles powered by AMS MathViewer

by Jun Shung Hwang
Trans. Amer. Math. Soc. 276 (1983), 335-346 Request permission

## 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.
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