I'm trying to make an exercise in a complex analysis textbook, but I'm stuck, so I hope someone can help me out. The exercise is assigned in a chapter about the Mittag-Leffler theorem.

1) If $f$ is an holomorphic function on $\mathbb{C}$, with exception of a finite amount of poles and $f$ is bounded by a polynomial for $\vert z \vert \ge R$, then show that $f$ is a rational function.

I think I succeeded in proving this part, by subtracting the singularities from $f$ and using (a modified version) of Liouville theorem. I mention the question because it may be related to the following questions.

2) If $f$:$\mathbb{C}$\ {$z_1$,...,$z_n$} $\rightarrow$$\mathbb{C}$ is injective, continuous and not bounded, then show that $\lim_{z \rightarrow \infty}f^{-1}(z) \in \{z_1,...,z_N,\infty\}$. (Hint: Weierstrass)

3)If $f:\mathbb{C}$\ $\{$$z_1$$,...,$$z_n$$ \}$ $\rightarrow \mathbb{C}$\ $\{$$w_1$$,...,$$w_n$$\}$ is a holomorphic bijection, with $z_1,...,z_n$ poles of $f$, then show that $f$ is a rational function.

I don't really see how to solve 2) and 3).


1 Answer 1


For (2) as stated, if $\lim_{n \to \infty} |x_n| = \infty$, then any convergent subsequence of $[f^{-1}(x_n)]_{n \in N}$ could not converge to anything in $C$\ $\{ z_1,...,z_n \}$ else $f$ would be discontinuous. Were there more conditions on $f$ ?


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