On the Boundary Behaviour of Univalent Harmonic Mappings onto Convex Domains

Bshouty, Daoud; Lyzzaik, Abdallah; Weitsman, Allen

doi:10.1007/bf03321687

Cited by 5 publications

(6 citation statements)

References 7 publications

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“…Because the dilatation ! n .z/ equals g 0 n .z/= h 0 n .z/ D z 2n 2 , we know that f n maps arcs of ‫ބ@‬ to either concave arcs or to stationary points [Bshouty and Hengartner 1997;Bshouty et al 2008]. Letting z D e iÂ 2 ‫,ބ@‬ we see that the latter situation occurs.…”

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confidence: 87%

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2013

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The singly periodic Scherk surfaces with higher dihedral symmetry have 2n-ends that come together based upon the value of '. These surfaces are embedded provided that 2 n < n 1 n ' < 2. Previously, this inequality has been proved by turning the problem into a Plateau problem and solving, and by using the Jenkins-Serrin solution and Krust's theorem. In this paper we provide a proof of the embeddedness of these surfaces by using some results about univalent planar harmonic mappings from geometric function theory. This approach is more direct and explicit, and it may provide an alternate way to prove embeddedness for some complicated minimal surfaces.

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confidence: 87%

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2013

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“…In the last part of the paper, we present a generalized version of a theorem proved by Bshouty, Lyzzaik and Weitsman (see [3,Theorem 6]). There, we define Jordan curves in D, which end at a point ζ ∈ ∂D.…”

Section: Theorem Cmentioning

confidence: 99%

“…More precisely, we first give results analogous to classical Koebe's lemma. In the last part, we present a new approach and a generalized version of [3,Theorem 6].…”

Section: Introductionmentioning

confidence: 99%

Koebe and Caratheódory type boundary behavior results for harmonic mappings

Bshouty¹,

Chen²,

Evdoridis³

et al. 2020

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We study the behavior of the boundary function of a harmonic mapping from global and local points of view. Results related to the Koebe lemma are proved, as well as a generalization of a boundary behavior theorem by Bshouty, Lyzzaik and Weitsman. We also discuss this result from a different point of view, from which a relation between the boundary behavior of the dilatation at a boundary point and the continuity of the boundary function of our mapping can be seen.

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“…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.…”

Section: Theorem 5 Suppose the Followingmentioning

confidence: 99%

“…In [4] it is observed that certain Möbius transformations on an inner dilatation transform it into a Blaschke product, which results in another GRM of D onto a convex domain Ω with a Blaschke dilatation.…”

Section: Introductionmentioning

confidence: 99%