On a result by Clunie and Sheil-Small

Dariusz Partyka; Ken-ichi Sakan

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 66 (2012), / Harvested from The Polish Digital Mathematics Library

Access to full text
Full (PDF)

Résumé

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 $| G (z_{2}) - G (z_{1}) | < | H (z_{2}) - H (z_{1}) |$ holds for all distinct points $z_{1}, z_{2} \in 𝔻$ . Here $H$ and $G$ are holomorphic mappings in $𝔻$ determined by $F = H + \bar{G}$ , 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 $Ω$ .

Publié le : 2012-01-01
EUDML-ID : urn:eudml:doc:289728

@article{bwmeta1.element.ojs-doi-10_17951_a_2012_66_2_81-92,
     author = {Dariusz Partyka and Ken-ichi Sakan},
     title = {On a result by Clunie and Sheil-Small},
     journal = {Annales Universitatis Mariae Curie-Sk\l odowska, sectio A -- Mathematica},
     volume = {66},
     year = {2012},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2012_66_2_81-92}
}

Dariusz Partyka; Ken-ichi Sakan. On a result by Clunie and Sheil-Small. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 66 (2012) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2012_66_2_81-92/

Bibliographie

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.