Abstract
The main ideas are taken from the excellent papers of Abu-Muhanna, Hengartner and Schober [Abu-Muhanna, Y. and Schober, G., 1987, Harmonic mappings onto convex domains. Canadian Journal of Mathematics, 39(6), 1489–1530; Hengartner, W. and Schober, G., 1986, Harmonic mappings with given dilatation. Journal of the London Mathematical Society, 33(2), 473–483]. However in practice, we need more general results on solutions of the partial differential equation , where ω is analytic in the open disk Δ, ω(Δ)⊂Δ, f (Δ)=D and D is a given simply connected domain, not necessary bounded or convex.
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