General Approach to Regions of Variability via Subordination of Harmonic Mappings

A planar harmonic mapping in a simply connected domain D ⊂ C is a complex-valued function f u iv defined in D for which both u and v are real harmonic in D, that is, Δf 4fzz 0, where Δ represents the Laplacian operator. The mapping f can be written as a sum of an analytic and antianalytic functions, that is, f h g. We refer to 1 and the book of Duren 2 for many interesting results on planar harmonic mappings. We note that the composition f ◦φ of a harmonic function f with an analytic function φ is harmonic, but this is not true for the function φ◦f , that is, an analytic function of a harmonic function need not be harmonic. It is known that 2, Theorem 2.4 the only univalent harmonic mappings of C onto C are the affine mappings g z βz γz η |β|/ |γ | . Motivated by the work of 3 , we say that F is an affine harmonic mapping of a harmonic mapping of f if and only if F has the form