On a result by Clunie and Sheil-Small
Dariusz Partyka; Ken-ichi Sakan
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2012)
- Volume: 66, Issue: 2
- ISSN: 0365-1029
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topDariusz Partyka, and Ken-ichi Sakan. "On a result by Clunie and Sheil-Small." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 66.2 (2012): null. <http://eudml.org/doc/289728>.
@article{DariuszPartyka2012,
abstract = {In 1984 J. Clunie and T. Sheil-Small proved ([2, Corollary 5.8]) that for any complex-valued and sense-preserving injective harmonic mapping $F$ in the unit disk $\mathbb \{D\}$, if $F(\mathbb \{D\})$ is a convex domain, then the inequality $|G(z_2)-G(z_1)| < |H(z_2)- H(z_1)|$ holds for all distinct points $z_1, z_2 \in \mathbb \{D\}$. Here $H$ and $G$ are holomorphic mappings in $\mathbb \{D\}$ determined by $F = H + \overline\{G\}$, up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain $\Omega $ in $\mathbb \{C\}$ and improve it provided $F$ is additionally a quasiconformal mapping in $\Omega $.},
author = {Dariusz Partyka, Ken-ichi Sakan},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Harmonic mappings; Lipschitz condition; bi-Lipchitz condition; co-Lipchitz condition; quasiconformal mappings},
language = {eng},
number = {2},
pages = {null},
title = {On a result by Clunie and Sheil-Small},
url = {http://eudml.org/doc/289728},
volume = {66},
year = {2012},
}
TY - JOUR
AU - Dariusz Partyka
AU - Ken-ichi Sakan
TI - On a result by Clunie and Sheil-Small
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2012
VL - 66
IS - 2
SP - null
AB - In 1984 J. Clunie and T. Sheil-Small proved ([2, Corollary 5.8]) that for any complex-valued and sense-preserving injective harmonic mapping $F$ in the unit disk $\mathbb {D}$, if $F(\mathbb {D})$ is a convex domain, then the inequality $|G(z_2)-G(z_1)| < |H(z_2)- H(z_1)|$ holds for all distinct points $z_1, z_2 \in \mathbb {D}$. Here $H$ and $G$ are holomorphic mappings in $\mathbb {D}$ determined by $F = H + \overline{G}$, up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain $\Omega $ in $\mathbb {C}$ and improve it provided $F$ is additionally a quasiconformal mapping in $\Omega $.
LA - eng
KW - Harmonic mappings; Lipschitz condition; bi-Lipchitz condition; co-Lipchitz condition; quasiconformal mappings
UR - http://eudml.org/doc/289728
ER -
References
top- Bshouty, D., Hengartner, W., Univalent harmonic mappings in the plane, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 48 (1994), 12-42.
- Clunie, J., Sheil-Small, T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 9 (1984), 3-25.
- Lewy, H., On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), 689-692.
- Partyka, D., The generalized Neumann-Poincare operator and its spectrum, Dissertationes Math., vol. 366, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 1997.
- Partyka, D., Sakan, K., A simple deformation of quasiconformal harmonic mappings in the unit disk, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 37 (2012), 539-556.
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