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

Abstract

Let ${\mathcal S}$ denote the set of all univalent analytic functions $f(z)=z+\sum _{n=2}^{\infty }a_n z^n$ on the unit disk $\mathbb {D}$. In 1946, B. Friedman found that the set $\mathcal S$ of those functions which have integer coefficients consists of only nine functions. In 1985, authors determine all functions in ${\mathcal S}$ 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 $f(z)=z+\sum _{n=2}^{\infty }a_n z^n+\sum _{n=1}^{\infty }\overline{b_n} \overline{z^n}$ with the coefficients $a_n$, $b_n$, $n=2, 3, \cdots$, in a fixed complex quadratic field $Q(\sqrt{d}i)$, where d is a positive square-free integer. Then, under the condition F is sense-preserving in $\mathbb {D}$ and $h'(z)\ne 0$ or F is sense-reversing in $\mathbb {D}$ and $g(z)\equiv 0$, we determine all univalent biharmonic mappings $F(z)=|z|^2(h(z)+\overline{g(z)})=|z|^2(z+\sum _{n=2}^{\infty }a_n z^n+\sum _{n=1}^{\infty }\overline{b_n} \overline{z^n})$, where $a_n, b_n\in Q(\sqrt{d}i)$, $n=2, 3, \cdots$.