Non-tangential limits of slowly growing analytic functions

Barth, Karl F. and Rippon, Philip J. (2008). Non-tangential limits of slowly growing analytic functions. Computational Methods and Function Theory, 8(1) pp. 85–99.

URL: http://www.heldermann.de/CMF/CMF08/CMF081/cmf08008...

Abstract

We show that if $f$ is an analytic function in the unit disc, $M(r,f) = {\rm O}((1-r)^{-\eta})$ as $r \to 1$, for every $\eta>0$, and $\sup_{0 \leq r < 1} (1-r)^s |f'(r \zeta)|<\infty$, where $|\zeta|=1,\,s<1,$ then $f$ has a finite non-tangential limit at $\zeta$. We also show that in this result it is not sufficient to assume that $M(r,f)= {\rm O}((1-r)^{-\eta})$ as $r \to 1$, for some fixed $\eta>0$.

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