Computational Methods and Function Theory 8 (2008), No. 2, 373--384
Copyright Heldermann Verlag 2008
Christian Pommerenke
pommeren@math.tu-berlin.de , Technische Universität Berlin, Institut für Mathematik, MA 8-2, 10623 Berlin, Germany.
Let $f$ be analytic and $f'(z)\ne 0$ in $\mathbb{D}$ and let $$ A_f(z) = \frac{1-|z|^2}{2} \frac{f''(z)}{f'(z)}-\overline{z} \qquad \mbox{for } z\in\mathbb{D}. $$ Many properties of $f$ can be described by the (linear-invariant) order $$ \sup_{z\in\mathbb{D}}|A_f(z)|. $$ The work of Avkhadiev and Wirths led to the introduction of the lower order of $f$ defined by $\inf_{z\in\mathbb{D}}|A_f(z)|$. It is perhaps a surprise that there are many (necessarily unbounded) functions of positive lower order. This paper studies some properties of these functions.
Keywords: Locally univalent, lower order, linear-invariant, Poincaré metric, trajectory.
MSC 2000: 30C99, 30C35, 30C45.
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