Computational Methods and Function Theory 8 (2008), No. 1, 85--99
Copyright Heldermann Verlag 2008
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.
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.
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