On harmonic Bloch-type mappings

Abstract

Let f be a complex-valued harmonic mapping defined in the unit disk . We introduce the following notion: we say that f is a Bloch-type function if its Jacobian satisfiesThis gives rise to a new class of functions which generalizes and contains the well-known analytic Bloch space. We give estimates for the schlicht radius, the growth and the coefficients of functions in this class. We establish an analogue of the theorem which, roughly speaking, states that for analytic is Bloch if and only if is univalent.