Let and 1 be Jordan domains, let µ ∈ (0, 1], and let f : → 1 be a harmonic homeomorphism. The object of the paper is to prove the following results: (a) If f is q.c. and ∂ , ∂ 1 ∈ C 1,µ , then f is Lipschitz; (b) if f is q.c., ∂ , ∂ 1 ∈ C 1,µ and 1 is convex, then f is bi-Lipschitz; and (c) if is the unit disk, 1 is convex, and ∂ 1 ∈ C 1,µ , then f is quasiconformal if and only if its boundary function is bi-Lipschitz and the Hilbert transform of its derivative is in L ∞ . These extend the results of Pavlović (Ann. Acad. Sci. Fenn. 27:365-372, 2002).

Abstract. Let QC(K, g) be a family of K-quasiconformal mappings of the open unit disk onto itself satisfying the PDE Δw = g, g ∈ C(U), w(0) = 0. It is proved that QC(K, g) is a uniformly Lipschitz family. Moreover, if |g| ∞ is small enough, then the family is uniformly bi-Lipschitz. The estimations are asymptotically sharp as K → 1 and |g| ∞ → 0, so w ∈ QC(K, g) behaves almost like a rotation for sufficiently small K and |g| ∞ .

Abstract. We study the Schwarz lemma for harmonic functions and prove sharp versions for the cases of real harmonic functions and the norm of harmonic mappings.

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Let H denote the class of all normalized complex-valued harmonic functions f = h + g in the unit disk D, and let S 0 H denote the class of univalent and sense-preserving functions f in H such that f z (0) = 0. If K = H + G denotes the harmonic Koebe function whose dilation isHere, a n , b n , A n , and B n denote the Maclaurin coefficients of h, g, H, and G. We show that the radius of univalence of the family F is 0.112903 . . .. We also show that this number is also the radius of the fully starlikeness of F . Analogous results are proved for a family which contains the class of harmonic convex functions in H. We use the new coefficient estimate for bounded harmonic mappings and Lemma 1.6 to improve Bloch-Landau constant for bounded harmonic mappings.