Criteria for univalence and quasiconformal extension of harmonic mappings in terms of the Schwarzian DERIVATIVE

Lemma 1. As before, let Rλ = maxω∈A0 λ |ω′(0)| . Then lim λ→0+ Rλ = 0 .

Theorem 1. There exists δ0 > 0 such that if ∥Sf∥≤δ0, then f is univalent.

Theorem 2. Let f be a sense-preserving harmonic mapping in the unit disk with ∥Sf∥≤δ0t for some t < 1, where δ0 is as in Theorem 1. Assume, in addition, that the dilatation ωf of f satisfies ∥ωf∥∞= sup z∈D |ωf(z)| < 1 . Then f can be extended to a quasiconformal map in bC. Before proving this second theorem, we would like to stress that the hypotheses ∥ω∥∞< 1 cannot be removed as the following example shows. Example. Consider the sense-preserving harmonic mapping f = z+g, where g′ equals the lens