Polynomial approximation of certain biharmonic mappings

In this paper we first determine conditions on the analytic polynomials pp and qq such that the family of univalent sense-preserving biharmonic polynomials of the form P=|z|2(p+q¯) is dense in the family of univalent sense-preserving biharmonic mappings FF defined on the unit disk DD of the form F=|z|2GF=|z|2G. Next, we consider the univalency of harmonic function GG whenever F=|z|2GF=|z|2G is univalent in DD. Finally, we give a partial answer to the problem raised by Muhanna about the radius of univalence of a family of sense-preserving biharmonic mappings with the form F=|z|2G+HF=|z|2G+H, where GG is univalent in DD.