On the generalized Seidel class 𝑈

Abstract

Abstract. As usual, we say that a function/ G U if /is meromorphic in | z | < 1 and has radial limits of modulus 1 a.e. (almost everywhere) on an arc A of | z \ = 1. This paper contains three main results: First, we extend our solution of A. J. Lohwater 's problem (1953) by showing that if / e U and/has a singular point P on A, and if v and \/v are a pair of values which are not in the range of /at P, then one of them is an asymptotic value of/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 / £ U and /(0) = 0, then the range of / over | z | < 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 | z | = 1, and preserves a sector between domain and range. We study the boundary behaviour and the representation of functions in this class.