A Remark on the Loewner Diierential Equation

We give a control-theoretic proof of Pommerenke's result on the parametric representation of normalized univalent functions in the unit disk as solutions of the Loewner diierential equation. The method consists in combining a classical result on nite-dimensional control-linear systems with Montel's theorem on normal families. x1 Introduction We rst recall some important subsets of the vector space H(D) of all functions analytic in the unit disk D = fz 2 C : jzj < 1g. Equipped with the topology of locally uniform convergence, H(D) becomes a metrizable locally convex Hausdorr space. For xed M 2 1; 1] let S M be the class of all functions f analytic and univalent in D satisfying jf(z)j < M (jzj < 1), normalized by f(0) = 0, f 0 (0) = 1. We put S := S 1. Let P denote the set of functions p that are analytic in D and satisfy Re p(z) > 0 (jzj < 1) and p(0) = 1. It is well-known that extr P = z 7 ! 1 + xz 1 ? xz jxj = 1 (1.1) is the set of extreme points of the class P. Deenition 1. Let I 0; 1] be an interval and H(D) a normal family. A function h : D I ! C is called a Carath eodory function on I with values in if the following conditions are satissed: (i) The mapping z 7 ! h(z; t) belongs to for almost all t 2 I. (ii) The mapping t 7 ! h(z; t) is measurable on I for each xed z 2 D. The set of all Carath eodory functions on I with values in will be denoted by L