Tag Archives: univalent

Some generalizations of theorems of holomorphic function.

\mathbf{Thm:} If f\not\equiv constant is analytic in a domain U, then f(U) is also a domain. If f only has poles in U, this conclusion still holds.

\mathbf{Thm:} f(z) holomorphic at z_0\in \overline{\mathbb{C}}, where it has a zero point of order k\geq 1, then there is a neighborhood of \mathcal{N}(z_0) and a neighborhood of \mathcal{N}(w_0) such that every point in \mathcal{N}(w_0) except w_0 itself has precisely k distinct inverse images in \mathcal{N}(z_0). If f has pole of order k at z_0, conclusion still holds.

\mathbf{Proof:} Suppose f has pole \phi(z)=\frac{1}{f(z)}, \eta=\frac 1 w. Observe that w=\frac {1}{\eta} carries |\eta|<\delta into the disc |w|>\frac 1\delta. If z_0=\infty, consider both transformations \xi=\frac 1z, \eta=\frac 1 w. \square

\mathbf{Corollary:} If w=f(z) is univalent(injective) on a domain G, then all the zeros and poles of f(z) in G  are simple.