ON HARMONIC QUASICONFORMAL MAPPINGS WITH FINITE AREA

In this paper, we study the class of harmonic K-quasiconformal mappings of the unit disk U with finite Euclidean areas vertical bar f(U)vertical bar(euc). We first give the Schwarz-pick lemma (cf. [8]) for this class of mappings as follows: vertical bar f(z)(z)vertical bar <= root vertical bar f(U)vertical bar(euc)/pi(1 - k(2)) 1/1 - vertical bar z vertical bar, z is an element of U, where k = K-1/K+1. Furthermore, we obtain the sharp coefficient- estimates of this class of mappings. As an application, for harmonic mappings f is an element of S-H(0) with finite vertical bar f(U)vertical bar(euc) we obtain sharp coefficient estimates.