Transactions of the American Mathematical Society

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

 

 

On the generalized Seidel class $ U$


Author: Jun Shung Hwang
Journal: Trans. Amer. Math. Soc. 276 (1983), 335-346
MSC: Primary 30C80
MathSciNet review: 684513
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Abstract: As usual, we say that a function $ f \in U$ if $ f$ is meromorphic in $ \vert z \vert < 1$ and has radial limits of modulus $ 1$ a.e. (almost everywhere) on an arc $ A$ of $ \left\vert z \right\vert = 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\vert z \right\vert < 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\vert z \right\vert = 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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DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0684513-8
Keywords: Boundary behaviour, generalized Seidel class, reflection principle, sector preserving function, star-like function
Article copyright: © Copyright 1983 American Mathematical Society