On existence of Becker extension

Abstract

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 $(f_t{)}_{t⩾0}$ such that the Herglotz function $p$ in the Loewner-Kufarev PDE

$\partial f_t(z)/\partial f=z{f}_t^{\prime }(z)p(z,t)$, $z\in \mathbb{D}$, a.e. $t\ge 0$,

satisfies $|(p(z,t)-1)/(p(z,t)+1)|\le k<1$, then $f$ admits a $k$-q.c. (= "$k$-quasiconformal") extension $F:\mathbb{C}\to \mathbb{C}$. 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 $k_{\ast }\in (0,1]$ with the property that for any $q\in (0,k_{\ast })$ there exists $k_0(q)\in (0,1)$ such that every normalized univalent function $f:\mathbb{D}\to \mathbb{C}$ with a $q$-q.c. extension to $\mathbb{C}$ satisfies Becker's condition with $k:=k_0(q)$. We prove that $k_{\ast }\ge 1/3$.