Boundary values versus dilatations of harmonic mappings

Curvature Properties of Planar Harmonic Mappings

“…This phenomenon is implicit in work of Hengartner and Schober [8]. An explicit proof, also based on Hopf's lemma, was given by Duren and Khavinson [6], and Bshouty and Hengartner [1] gave another proof, but the above argument is more direct. The result can be summarized as follows.…”

“…A formula is derived for the curvature of the local image of a circle |z| = r. When f has a smooth extension to an arc of the unit circle where its dilatation has unit modulus, the formula shows that the image arc is concave. This is a known result [8,6,1], but the natural proof based on curvature appears to be new. When the dilatation is a finite Blaschke product, the curvature formula leads to a kind of Gauss-Bonnet formula that relates total curvature of the boundary with angles at corners.…”

On the boundary behavior of univalent harmonic mappings

“…Note that Lemma 4 (a) and (c) follow from Theorem 2.2 and Corollary 2.8 respectively of [2], and Lemma 4 (b) is itself Corollary 2(i) of [4]. We divide the proof into two parts: A.…”

Uniqueness of harmonic mappings with Blaschke dilatations

“…For the exceptional values of λ = λ j , then n(Γ λ , 0) undergoes a jump decrease in both cases ii) and iii), then a compensating jump increase in case iii). Since n(Γ 0 , 0) = 1, equation (2) yields n(Γ λ , 0) = 1 for all 0 ≤ λ < 1.…”