Geodesic
On a result by Clunie and Sheil-Small
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 66 (2012) no. 2.

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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(z_2)-G(z_1)| lt; |H(z_2)- H(z_1)| holds for all distinct points z_1, z_2 ∈𝔻. Here H and G are holomorphic mappings in 𝔻 determined by F = H + 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.
Keywords: Harmonic mappings, Lipschitz condition, bi-Lipchitz condition, co-Lipchitz condition, quasiconformal mappings 
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Partyka, Dariusz; Sakan, Ken-ichi. On a result by Clunie and Sheil-Small. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 66 (2012) no. 2. http://geodesic.mathdoc.fr/item/AUM_2012_66_2_a5/

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