# On existence of Becker extension

## Authors

• Pavel Gumenyuk Politecnico di Milano, Department of Mathematics

## Keywords:

Univalent function, boundary behavior, quasiconformal extension, Loewner chain, 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 $$k_*\ge 1/3$$.

Issue
Section
Articles

2022-07-12

## How to Cite

Gumenyuk, P. (2022). On existence of Becker extension. Annales Fennici Mathematici, 47(2), 979–1005. https://doi.org/10.54330/afm.120591