On existence of Becker extension

نویسندگان

چکیده

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

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ژورنال

عنوان ژورنال: Annales Fennici Mathematici

سال: 2022

ISSN: ['2737-0690', '2737-114X']

DOI: https://doi.org/10.54330/afm.120591