Parametric representation of univalent functions with boundary regular fixed points

Abstract

Given a set $\mathfrak{S}$ of conformal maps of the unit disk $\mathbb{D}$ into itself that is closed under composition, we address the question whether $\mathfrak{S}$ can be represented as the reachable set of a Loewner - Kufarev - type ODE $\mathrm{d}{w}_t/\mathrm{d}t=G_t\circ w_t$, $w_0={\mathsf{id}}_{\mathbb{D}}$, where the control functions $t\mapsto G_t$ form a convex cone ${\mathcal{M}}_{\mathfrak{S}}$. For the set of all conformal $\phi :\mathbb{D}\to \mathbb{D}$ with $\phi (0)=0$, ${\phi }^{\prime }(0)>0$, an affirmative answer to this question is the essence of Loewner's well-known Parametric Representation of univalent functions [Math. Ann. 89 (1923), 103-121]. In this paper, we study classes of conformal self-maps defined by their boundary regular fixed points and, in part of the cases, establish their Loewner-type representability.

Version 3: Introduction has been modified; Example 3.1 added + other minor corrections