Univalent Harmonic and Biharmonic Mappings with Integer Coefficients in Complex Quadratic Fields

Let S denote the set of all univalent analytic functions on the unit disk . In 1946, B. Friedman found that the set of those functions which have integer coefficients consists of only nine functions. In 1985, authors determine all functions in whose coefficients are integers in complex quadratic fields. The first aim of this paper is to determine the class of univalent sense-preserving harmonic mappings with the coefficients , , , in a fixed complex quadratic field , where d is a positive square-free integer. Then, under the condition F is sense-preserving in and or F is sense-reversing in and , we determine all univalent biharmonic mappings , where a(n), b(n) is an element of Q(root di), n = 2,3, ....