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Geometric properties of harmonic shears. (English) Zbl 1056.30023
A study of planar harmonic mappings produced with the shear construction devised by Clunie and Sheil-Small. Specifically it will describe the geometry of mappings produced by the shear construction. A technique for constructing examples of harmonic mappings by shearing a conformal mapping is presented. Some examples of harmonic shears are illustrating graphically with the help of Mathematica software. For a given conformal mapping $$\varphi$$ convex in the horizontal direction in $$D$$ and analytic function $$\omega$$ with $$|\omega(z) |<1$$ for $$z\in D$$, the shear of $$\varphi$$ for dilatation $$\omega$$ is defined as $$f=h+\overline g$$ where $$h$$ and $$g$$ are analytic functions satisfying the pair of differential equations $$h'-g'=\varphi'$$ $$\omega h'-g'-0$$ with normalization $$h(0)=\varphi(0)$$, $$g(0)=0$$. Without loss of generality it may be assumed that $$\varphi(0)=0$$, and then $f(z)=\text{Re} \left( \int^z_0\frac{2 \varphi'(\zeta)} {1-\omega(\zeta)}d \zeta\right)-\overline {\varphi(z)}.$ The following result is also proved: Theorem 1. Let $$\varphi$$ be a conformal mapping of the unit disc $$D$$ onto a domain convex in the direction of the real axis, and let $$\omega$$ be an analytic function with $$|\omega(z) |<1$$ in $$D$$. Let $$f$$ be the horizontal shear $$\varphi$$ with dilatation $$\omega$$. If $$I=\{e^{i\theta}: \theta\in (a,b)\}$$ is an arc which $$\varphi$$ maps to a horizontal line segment and if $$\omega$$ is continuous up to $$I$$ with $$\omega(e^{i\theta})\in T \setminus\{1\}= \{e^{i\theta}:\theta \in(0,2\pi)\}$$ for all $$\theta\in (a,b)$$, then the image of the arc $$I$$ under $$f$$ collapses to a single point in the sense that $$f$$ extends continuously to $$I$$ with $$f(e^{i\theta})= \lim_{z\to e^{i\theta}} f(z)=c_0$$ for some constant $$c_0$$ and for all $$\theta\in(a,b)$$.

##### MSC:
 30C99 Geometric function theory 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 30C55 General theory of univalent and multivalent functions of one complex variable
##### Keywords:
harmonic mapping; harmonic shear; boundary behavior
Mathematica
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##### References:
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