On a result by Clunie and Sheil-Small

Dariusz Partyka; Ken-ichi Sakan

Annales UMCS, Mathematica (2012)

  • Volume: 66, Issue: 2, page 81-92
  • ISSN: 2083-7402

Abstract

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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 D, if F(D) is a convex domain, then the inequality |G(z2)− G(z1)| < |H(z2) − H(z1)| holds for all distinct points z1, z2∈ D. Here H and G are holomorphic mappings in D determined by F = H + Ḡ, up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain Ω in ℂ and improve it provided F is additionally a quasiconformal mapping in Ω.

How to cite

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Dariusz Partyka, and Ken-ichi Sakan. "On a result by Clunie and Sheil-Small." Annales UMCS, Mathematica 66.2 (2012): 81-92. <http://eudml.org/doc/267950>.

@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 D, if F(D) is a convex domain, then the inequality |G(z2)− G(z1)| < |H(z2) − H(z1)| holds for all distinct points z1, z2∈ D. Here H and G are holomorphic mappings in D determined by F = H + Ḡ, up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain Ω in ℂ and improve it provided F is additionally a quasiconformal mapping in Ω.},
author = {Dariusz Partyka, Ken-ichi Sakan},
journal = {Annales UMCS, Mathematica},
keywords = {harmonic mappings; Lipschitz condition; bi-Lipschitz condition; co-Lipschitz condition; quasiconformal mappings},
language = {eng},
number = {2},
pages = {81-92},
title = {On a result by Clunie and Sheil-Small},
url = {http://eudml.org/doc/267950},
volume = {66},
year = {2012},
}

TY - JOUR
AU - Dariusz Partyka
AU - Ken-ichi Sakan
TI - On a result by Clunie and Sheil-Small
JO - Annales UMCS, Mathematica
PY - 2012
VL - 66
IS - 2
SP - 81
EP - 92
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 D, if F(D) is a convex domain, then the inequality |G(z2)− G(z1)| < |H(z2) − H(z1)| holds for all distinct points z1, z2∈ D. Here H and G are holomorphic mappings in D determined by F = H + Ḡ, up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain Ω in ℂ and improve it provided F is additionally a quasiconformal mapping in Ω.
LA - eng
KW - harmonic mappings; Lipschitz condition; bi-Lipschitz condition; co-Lipschitz condition; quasiconformal mappings
UR - http://eudml.org/doc/267950
ER -

References

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  1. [1] Bshouty, D., Hengartner, W., Univalent harmonic mappings in the plane, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 48 (1994), 12-42. Zbl0894.30014
  2. [2] Clunie, J., Sheil-Small, T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 9 (1984), 3-25. Zbl0506.30007
  3. [3] Lewy, H., On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), 689-692. Zbl0015.15903
  4. [4] Partyka, D., The generalized Neumann-Poincaré operator and its spectrum, Dissertationes Math., vol. 366, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 1997. Zbl0885.30014
  5. [5] Partyka, D., Sakan, K., A simple deformation of quasiconformal harmonic mappingsin the unit disk, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 37 (2012), 539-556. Zbl1272.30035

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