On the Radius of Starlikeness for Harmonic Mappings

Bagapsh, A. O.

doi:10.1007/s10958-018-4003-x

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“…This implies that if β ∈ (1, 2), then the dependence ϕ(θ), where θ is the exit angle of the ray Λ θ leaving the preimage w = 0 of the corner vertex, while ϕ is the exit angle of the image L θ = F −1 (Λ θ ) of the ray, leaving the corner vertex z = 0 itself, is described by the relation Figure 4 displays the graph of dependence (23). Figure 3c displays F −1 (Ξ) on the plane z, i. e., the image under the inverse map for the polar mesh defined by (20) (for case II).…”

Section: The Map Inverse To the Harmonic One: Behavior Near The Preimmentioning

confidence: 99%

“…The interest to this direction is caused by its great theoretical importance. In particular, harmonic maps play a substantial role in differential geometry and the theory of minimal surfaces (see [13], [14]), in the geometric function theory (see [15]- [20]), and in the geometric theory of quasiconformal maps (see [21]- [29]), including the extension problem of the Riemann theorem for those maps (see [3], [30]- [33]).…”

Section: Introductionmentioning

confidence: 99%

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Singular behavior of harmonic maps near corners

Безродных

Власов

2019

Complex Variables and Elliptic Equations

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For a harmonic map F : Z harm −→ W transforming the contour of a corner of the boundary ∂Z into a rectilinear segment of the boundary ∂W, the behavior near the vertex of the specified corner is investigated. The behavior of the inverse map F −1 : W −→ Z near the preimage of the vertex is investigated as well. In particular, we prove that if ϕ is the value of the exit angle from the vertex of the reentrant corner for a smooth curve L and θ is the value of the exit angle from the vertex image for the image F(L) of the specified curve, then the dependence of θ on ϕ is described by a discontinuous function. Thus, such a behavior of the harmonic map qualitatively differs from the behavior of the corresponding conformal map: for the latter one, the dependence θ(ϕ) is described by a linear function.

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Section: The Map Inverse To the Harmonic One: Behavior Near The Preimmentioning

confidence: 99%