Parametric representation of univalent functions with boundary regular fixed points

TL;DR

This paper investigates whether certain classes of conformal self-maps of the unit disk, characterized by boundary fixed points, can be represented through Loewner-type differential equations, extending classical results in univalent function theory.

Contribution

It extends Loewner's parametric representation to classes of conformal maps with boundary regular fixed points, establishing their Loewner-type representability.

Findings

Confirmed Loewner-type representation for specific classes of boundary fixed point maps.

Extended classical Loewner theory to new classes of conformal self-maps.

Provided conditions under which these classes are reachable via Loewner equations.

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 $\varphi:\mathbb D\to\mathbb D$ with $\varphi(0)=0$, $\varphi'(0)>0$, 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.