On existence of Becker extension

Pavel Gumenyuk¹

¹Politecnico di Milano, Department of Mathematics

Annales Fennici Mathematici, Tome 47 (2022) no. 2, pp. 979-1005

Résumé

A well-known theorem by Becker states that if a normalized univalent function $f$ in the unit disk $\mathbb{D}$ can be embedded as the initial element into a Loewner chain $(f_t)_{t\geqslant 0}$ such that the Herglotz function $p$ in the Loewner-Kufarev PDE $\partial f_t(z)/\partial f=zf'_t(z)p(z,t)$, $z\in\mathbb{D}$, a.e. $t\ge0$,satisfies $\big|(p(z,t)-1)/(p(z,t)+1)\big|\le k<1$, then $f$ admits a $k$-q.c. (= "$k$-quasiconformal") extension $F\colon\mathbb{C}\to\mathbb{C}$. The converse is not true. However, a simple argument shows that if $f$ has a $q$-q.c. extension with $q\in(0,1/6)$, then Becker's condition holds with $k:=6q$. In this paper we address the following problem: find the largest $k_*\in(0,1]$ with the property that for any $q\in(0,k_*)$ there exists $k_0(q)\in(0,1)$ such that every normalized univalent function $f\colon\mathbb D\to\mathbb C$ with a $q$-q.c. extension to $\mathbb C$ satisfies Becker's condition with $k:=k_0(q)$. We prove that $k_*\ge 1/3$.  

Détail

DOI : 10.54330/afm.120591

Keywords: Univalent function, boundary behavior, quasiconformal extension, Loewner chain, Becker extension