Univalent function

In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective.

Any mapping ${\displaystyle \phi _{a}}$ of the open unit disc to itself. The function

where ${\displaystyle |a|<1,}$ is univalent.

One can prove that if ${\displaystyle G}$ and ${\displaystyle \Omega }$ are two open connected sets in the complex plane, and

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