A decomposition theorem for planar harmonic mappings

A necessary and sufficient condition is found for a complex-valued harmonic function to be decomposable as an analytic function followed by a univalent harmonic mapping. In the theory of quasiconformal mappings, it is proved that for any measurable function 1a with 11,uII ,, 0 there. According to a theorem of Lewy [3], the Jacobian of a univalent harmonic map in the plane can never vanish, so J(z) > 0 for sense-preserving univalent harmonic maps. It is instructive to begin with two simple examples. Example 1. Let f be the harmonic polynomial f(z) = z2 +3z Then f has dilatation a(z) = z, and f is sense-preserving in the unit disk ID = {z IzI < 1}. We claim that f has no decomposition of the desired form in any neighborhood of the origin. Suppose on the contrary that f = F o p, where (p is analytic near the origin and F is harmonic and univalent on the range of p. Then F is sensepreserving because f is. We may suppose without loss of generality that p(0) = 0. Received by the editors October 10, 1994. 1991 Mathematics Subject Classification. Primary 30C99; Secondary 31A05, 30C65.