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 \((f_t)_{t\geqslant 0}\) 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 \(k_*\ge 1/3\).