Angular derivatives of quasiconformal harmonic maps on the Poincaré disk

Angular derivatives of quasiconformal harmonic maps on the Poincaré disk Let $$S^1$$ S 1 be the boundary of the open unit disk $$\mathbb {D}$$ D and let h be a quasisymmetric homeomorphism from the unit circle $$S^1$$ S 1 onto itself. Let H be the quasiconformal harmonic extension of h to $$\mathbb {D}$$ D with respect to the Poincaré  metric. In this paper, it is shown that, if $$h'(\zeta )=\alpha \ne 0$$ h ′ ( ζ ) = α ≠ 0 at $$\zeta $$ ζ in $$S^1$$ S 1 , then when $$z\rightarrow \zeta $$ z → ζ in $$\mathbb {D}$$ D non-tangentially, $$\begin{aligned} \lim _{z\rightarrow \zeta }\frac{H(z)-H(\zeta )}{z-\zeta }=\alpha \end{aligned}$$ lim z → ζ H ( z ) - H ( ζ ) z - ζ = α and the complex derivatives $$H_z(z)$$ H z ( z ) and $$H_{\bar{z}}(z)$$ H z ¯ ( z ) approach $$\alpha $$ α and 0 respectively, i.e., H has an angular derivative $$\alpha $$ α at $$\zeta $$ ζ ; conversely, if H has a non-tangential derivative $$\alpha \ne 0$$ α ≠ 0 at $$\zeta $$ ζ , then $$h'(\zeta )=\alpha $$ h ′ ( ζ ) = α and hence H has an angular derivative $$\alpha $$ α at $$\zeta $$ ζ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Angular derivatives of quasiconformal harmonic maps on the Poincaré disk

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
D.O.I.
10.1007/s00209-017-1880-5
Publisher site
See Article on Publisher Site

Abstract

Let $$S^1$$ S 1 be the boundary of the open unit disk $$\mathbb {D}$$ D and let h be a quasisymmetric homeomorphism from the unit circle $$S^1$$ S 1 onto itself. Let H be the quasiconformal harmonic extension of h to $$\mathbb {D}$$ D with respect to the Poincaré  metric. In this paper, it is shown that, if $$h'(\zeta )=\alpha \ne 0$$ h ′ ( ζ ) = α ≠ 0 at $$\zeta $$ ζ in $$S^1$$ S 1 , then when $$z\rightarrow \zeta $$ z → ζ in $$\mathbb {D}$$ D non-tangentially, $$\begin{aligned} \lim _{z\rightarrow \zeta }\frac{H(z)-H(\zeta )}{z-\zeta }=\alpha \end{aligned}$$ lim z → ζ H ( z ) - H ( ζ ) z - ζ = α and the complex derivatives $$H_z(z)$$ H z ( z ) and $$H_{\bar{z}}(z)$$ H z ¯ ( z ) approach $$\alpha $$ α and 0 respectively, i.e., H has an angular derivative $$\alpha $$ α at $$\zeta $$ ζ ; conversely, if H has a non-tangential derivative $$\alpha \ne 0$$ α ≠ 0 at $$\zeta $$ ζ , then $$h'(\zeta )=\alpha $$ h ′ ( ζ ) = α and hence H has an angular derivative $$\alpha $$ α at $$\zeta $$ ζ .

Journal

Mathematische ZeitschriftSpringer Journals

Published: Mar 24, 2017

References

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