Harmonic automorphisms of the unit disk

Let H be the class of harmonic automorphisms of the unit disk D. The function F=h−g associated with f=h+ḡ∈H maps D conformally onto a horizontally convex domain Ω. Conversely, given Ω both f∈H and F with F(D)=Ω can be retrieved (Theorem 1). Compact subclasses H(M)⊂H consisting of Poisson extensions of M-quasisymmetric automorphisms of ∂D span H (Lemma 1). For f(reit)=∑n=−∞+∞cnr|n|eint∈H(M) the bounds of |cn| (upper one for n=0,2, lower one for n=1) and ∑n=−∞+∞|cn| are given (Theorems 2–4).