© 1992 by London Mathematical Society
Non-Tangential Limits for Analytic Functions of Slow Growth in a Disc
Análisis Matemático, Facultad de Ciencias, Universidad de Málaga 29071 Málaga, Spain
A classical theorem of Lindelöf asserts that if f is a function analytic and bounded in the unit disc U which has the asymptotic value L at a point 
then it has the non-tangential limit L at 
. This result cannot be extended to functions f analytic in U whose maximum modulus grows to infinity arbitrarily slowly. However, we obtain two results of Lindelöf type valid for these functions. We prove that if f is a function analytic in U such that sup
and 
> 0,
then f has a non-tangential limit at 
if it satisfies one of the conditions sup or there exists a curve 
that ends at and lies otherwise in U such that f' is bounded on .