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.
@article{bwmeta1.element.doi-10_2478_v10062-012-0015-z,
author = {Dariusz Partyka and Ken-ichi Sakan},
title = {On a result by Clunie and Sheil-Small},
journal = {Annales UMCS, Mathematica},
volume = {66},
year = {2012},
pages = {81-92},
zbl = {1307.31003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10062-012-0015-z}
}
Dariusz Partyka; Ken-ichi Sakan. On a result by Clunie and Sheil-Small. Annales UMCS, Mathematica, Tome 66 (2012) pp. 81-92. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10062-012-0015-z/
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