Fully Starlike and Convex Harmonic Mappings of order α
Abstract
The hereditary property of convexity and starlikeness for conformal mappings does not generalize to univalent harmonic mappings. This failure leads us to the notion of fully starlike and convex mappings of order α, (0≤ α<1). A bound for the radius of fully starlikeness and fully convexity of order α is determined for certain families of univalent harmonic mappings. Convexity is not preserved under the convolution of univalent harmonic convex mappings, unlike in the analytic case. Given two univalent harmonic convex mappings f and g, the problem of finding the radius r0 such that f*g is a univalent harmonic convex mapping in |z|<r0, is being considered.
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