Theorem 1 (Bloch)
There exists a positive constant
![$b$](img6.gif)
such that if
![$f\in H(B)$](img7.gif)
and
![$f^\prime(0)\neq
0$](img8.gif)
, then
![$f$](img9.gif)
maps some subdomain of B biholomorphically onto a
disc of radius
![$b\cdot\vert f^\prime(0)\vert$](img10.gif)
.
Definition 1 (Wu)
Let
![${B}^{n}$](img35.gif)
be the open unit ball of
![$\mathbb {C}^{n}$](img16.gif)
and let
![$\mathcal{F}:{B}^{n}\longrightarrow\mathbb {C}^{n}$](img36.gif)
be a family of
holomorphic mappings. We say
![$\mathcal{F}$](img37.gif)
is K-quasiregular iff there exists
a constant
![$K$](img27.gif)
so that, for each
![$f=(f_{1},\ldots,f_{n})$](img38.gif)
of
![$\mathcal{F}$](img37.gif)
, the
following holds throughout
![${B}^{n}$](img35.gif)
,
Theorem 2 (Wu, 1967)
Let
![$\mathcal{F}:{B}^{n}\longrightarrow\mathbb {C}^{n}$](img36.gif)
be a K-quasiregular family of
holomorphic mappings such that
![$\vert det \mathcal{J}_{f}$](img40.gif)
(0)
![$\vert$](img34.gif)
=1 for all
![$\in\mathcal{F}$](img41.gif)
.
Then, there is a positive constant
![$c$](img42.gif)
such that every
![$f\in\mathcal{F}$](img43.gif)
possesses a univalent ball
of radius
![$c$](img42.gif)
.