On the Schwarz reflection principle

Abstract:

Recently, we have solved a long outstanding problem of A. J. Lohwater (1953) by showing that if $f(z)$ is meromorphic in $|z|< 1$ whose radial limits have modulus 1 for almost all points on an arc $A$ of $|z|< 1$, and if $P$ is a singular point of $f(z)$ on $A$, then every value of modulus 1 which is not in the range of $f(z)$ at $P$ is an asymptotic value of $f(z)$ at some point of each subarc of $A$ containing the point $P$. Lohwater proved this theorem for functions of bounded characteristic and he made a comment that his method is not, in general, applicable to functions of unbounded characteristic. In this paper, we shall present an alternative proof of the above theorem based on the very method of Lohwater.