Abstract
We prove that if the Schwarzian norm of a given complex-valued locally univalent harmonic mapping $f$ in the unit disk is small enough, then $f$ is, indeed, globally univalent and can be extended to a quasiconformal mapping in the extended complex plane.
Results & Lemmas (3)
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Lemma 1.
Lemma 1. As before, let Rλ = maxω∈A0 λ |ω′(0)|. Then lim λ→0+ Rλ = 0.
Lemma 1. As before, let Rλ = maxω∈A0 λ |ω′(0)| . Then lim λ→0+ Rλ = 0 .
Theorem 1.
Theorem 1. There exists δ0 > 0 such that if ∥Sf∥≤δ0, then f is univalent.
Theorem 1. There exists δ0 > 0 such that if ∥Sf∥≤δ0, then f is univalent.
Theorem 2.
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,…
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
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