2020/03/23 by Pavel Gumenyuk, Gumenyuk, Pavel
Mathematics · Medicine · #Analytic and geometric function theory #Meromorphic and Entire Functions #Pharmacological Effects of Medicinal Plants
paper · pdf · doi:10.48550/arxiv.2003.10037
A well-known theorem by J. Becker states that if a normalized univalent function f in the unit disk \mathbbD can be embedded as the initial element into a Loewner chain (ft)t\geqslant 0 such that the Herglotz function p in the Loewner -- Kufarev PDE ∂ ft(z)/∂ f=zf't(z)p(z,t), z∈\mathbbD,\quada.e.~t≥0, satisfies |(p(z,t)-1)/(p(z,t)+1)|≤ k<1, then f admits a k-q.c. (="k-quasiconformal") extension F:ℂ→ℂ. The converse is not true. However, a simple argument shows that if f has a q-q.c. extension with q∈(0,1/6), then Becker's condition holds with k:=6q. In this paper we address the following problem: find the largest k_*∈(0,1] with the property that for any q∈(0,k_*) there exists k0(q)∈(0,1) such that every normalized univalent function f:\mathbb D→\mathbb C with a q-q.c. extension to \mathbb C satisfies Becker's condition with k:=k0(q). We prove that k_*≥1/3.