Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Next Article Computational Methods and Function Theory 8 (2008), No. 1, 85--99 Copyright Heldermann Verlag 2008 Non-Tangential Limits of Slowly Growing Analytic Functions Karl F. Barth kfbarth@mailbox.syr.edu , Syracuse University, Department of Mathematics, Syracuse NY 13244, U.S.A. Philip J. Rippon p.j.rippon@open.ac.uk , The Open University, Department of Mathematics, Walton Hall, Milton Keynes MK7 6AA, U.K. [Abstract-pdf] [Abstract-ps] We show that if $f$ is an analytic function in the unit disc $\mathbb{D}$, $M(r,f) = \LandauO((1-r)^{-\eta}) \qquad\mbox{as } r \to 1, \quad\mbox{for every } \eta>0,$ and $\sup_{0 \leq r < 1} (1-r)^s |f'(r \zeta)|<\infty, \qquad\mbox{where }|\zeta|=1,\,s<1,$ then $f$ has a finite non-tangential limit at $\zeta$. We also show that in this result it is not sufficient to assume that $M(r,f)= \LandauO((1-r)^{-\eta}) \qquad\mbox{as } r \to 1, \quad\mbox{for some fixed } \eta>0.$ Keywords: Non-tangential limit, Fatou point, slowly growing analytic function. MSC 2000: 30D40. [FullText-pdf (304 K)] [FullText-ps (504 K)]