Non-Tangential Limits of Slowly Growing Analytic Functions

We show that if f is an analytic function in the unit disc \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{D}$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\left( {r,f} \right) = \mathcal{O}\left( {\left( {1 - r} \right)^{ - \eta } } \right) as r \to 1, for every \eta > 0,$$\end{document}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathop {\sup }\limits_{0 \leqslant r < 1} (1 - r)^s \left| {f'\left( {r\zeta } \right) < \infty } \right|, where \left| \zeta \right| = 1, s < 1,$$\end{document} then f has a finite non-tangential limit at ζ. We also show that in this result it is not sufficient to assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\left( {r,f} \right) = \mathcal{O}\left( {\left( {1 - r} \right)^{ - \eta } } \right) as r \to 1, for some fixed \eta > 0.$$\end{document}