On harmonic Bloch-type mappings

Let $f$ be a complex-valued harmonic mapping defined in the unit disk $\mathbb D$. We introduce the following notion: we say that $f$ is a Bloch-type function if its Jacobian satisfies $$\sup_{z\in\mathbb D}(1-|z|^2)\sqrt{|J_f(z)|}<\infty.$$ This 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 states that an analytic $\varphi$ is Bloch if and only if there exists $c>0$ and a univalent $\psi$ such that $\varphi = c \log \psi'$.Comment: 11 pages, LaTeX; corrected typos in version 2; to appear in Complex Variables and Elliptic Equation

On harmonic Bloch-type mappings

Let f be a complex-valued harmonic mapping defined in the unit disk (Formula presented.). We introduce the following notion: we say that f is a Bloch-type function if its Jacobian satisfies (Formula presented.) This 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 ϕ analytic ϕ' is Bloch if and only if ϕ is univalen

On harmonic Bloch-type mappings

Let f be a complex-valued harmonicmapping defined in the unit disk D. We introduce the following notion: we say that f is a Bloch-type function if its Jacobian satisfies This 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 log. is Bloch if and only if. is univalent