TY - JOUR
AU - Gumenyuk, Pavel
PY - 2022/07/12
Y2 - 2022/08/17
TI - On existence of Becker extension
JF - Annales Fennici Mathematici
JA - Ann. Fenn. Math.
VL - 47
IS - 2
SE - Articles
DO - 10.54330/afm.120591
UR - https://afm.journal.fi/article/view/120591
SP - 979-1005
AB - <pre>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 <br><br>\(\partial f_t(z)/\partial f=zf'_t(z)p(z,t)\), \(z\in\mathbb{D}\), a.e. \(t\ge0\),<br><br>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 \(k_*\ge 1/3\).</pre><p> </p>
ER -