Results on bi-univalent functions

Abstract:

When the class $\sigma$ of bi-univalent functions was first defined, it was known that functions of the form $\phi \circ {\psi ^{ - 1}} \in \sigma$ when $\phi$ and $\psi$ are univalent, map the unit disc ${\mathbf {B}}$ onto a set containing ${\mathbf {B}}$, and satisfy $\phi (0) = \psi (0) = 0$, $\phi ’(0) = \psi ’(0)$. It is shown here that such functions form a proper subset of $\sigma$, and that $\sigma$ is a proper subset of the set of functions of the form $\phi \circ {\psi ^{ - 1}}$, where $\phi$ and $\psi$ are locally univalent, at most $2$-valent, each maps a subregion of ${\mathbf {B}}$ univalently onto ${\mathbf {B}}$, and $\phi (0) = \psi (0) = 0$, $\phi ’(0) = \psi ’(0)$, ${\psi ^{ - 1}}(0) = 0$. It is also shown that there are $f(z) = z + {a_2}{z^2} + \cdots$ in $\sigma$ with $\left | {{a_2}} \right | > 4/3$. However, doubt is cast that $\left | {{a_2}} \right |$ can be as large as $3/2$.