Locally one-to-one harmonic functions with starlike analytic part

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Let Lₕ denote the set of all normalized locally one-to-one and sense-preserving harmonic functions in the unit disc Δ. It is well-known that every complex-valued harmonic function in the unit disc Δ can be uniquely represented as f = h + g, where h and g are analytic in Δ. In particular the decomposition formula holds true for functions of the class Lₕ. For a fixed analytic function h, an interesting problem arises - to describe all functions g, such that f belongs to Lₕ. The case when f∈ Lₕ and h is the identity mapping was considered [2]. More general results are given in [3], where f∈ Lₕ and letting h to be a convex analytic mapping. The focus of our present research is to characterize the set of all functions f∈ Lₕ having starlike analytic part h. In this paper, we provide coefficient, distortion and growth estimates of g. We also give growth and Jacobian estimates of f.