On existence of Becker extension

A well-known theorem by Becker states that if a normalized univalent function f in the unit disk D can be embedded as the initial element into a Loewner chain (f(t))(t >= 0 )such that the Herglotz function p in the Loewner-Kufarev PDE partial derivative f(t)(z)/partial derivative f = zf(t)'(z)p(z, t), z is an element of D, a.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 is an element of (0, 1/6), then Becker's condition holds with k := 6q. In this paper we address the following problem: find the largest k(*) is an element of (0, 1] with the property that for any q is an element of (0, k(*)) there exists k(0)(q) is an element of (0, 1) such that every normalized univalent function f : D -> C with a q-q.c. extension to C satisfies Becker's condition with k := k(0)(q). We prove that k(*) >= 1/3.