17 questions linked to/from Entire one-to-one functions are linear
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### Relation between linearity and injectivity of an entire function [duplicate]

Given $f$ entire function on $\mathbb C$ and $f$ one-one. Is it true that $f$ is linear? At least among polynomials the only such functions are linear!
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### Suppose $f$ is entire and one-to-one. Show that $f(z)=az+b$. [duplicate]

My instinct is that it may be proved by Liouville's extended theorem. But how to do so? Or are there any other methods? Thanks!
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### An entire and one-to-one function must be of the form AZ+B, A non-zero. How to rule out higher degree polynomials in z? [duplicate]

Show that if f is entire and one-to-one, then it must be of the form AZ+B, with A not equal to zero. I am editing my question, since there are duplicates on this forum to the question of why an ...
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### one-to-one holomorphic map of $\mathbb{C}$ onto itself must be of the form $az+b$? [duplicate]

If $h: \mathbb{C} \rightarrow \mathbb{C}$ is a one-to-one holomorphic map, then $h(z)=az+b$, where $a,b\in\mathbb{C}$. How to prove this argument?
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### entire bijection of $\mathbb{C}$ with 2 fixed points

Besides the identity map, is there an entire function $f$ that is a bijection from $\mathbb{C}$ to $\mathbb{C}$ and has 2 fixed points? Thank you for the help.
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### Why can't an analytic function be injective in a neighborhood of an essential singularity?

Let $D \subset \mathbb{C}$ be a domain and let $a \in D$. Suppose $f: D \smallsetminus \{a\} \to \mathbb{C}$ is analytic and that $a$ is an essential singularity of $f$. Show that $f$ cannot be ...
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### Pole of order $\ge 2 \; \Rightarrow \;$ not injective

Let $D \subseteq \mathbb{C}$ be open and $f : D \rightarrow \mathbb{C}$ meromorphic with a pole of order $\ge 2$ in $a \in D$. Then $f$ is not injective. Is there an easy proof to this? This is not ...
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### Why is every holomorphic bijection of the Riemann sphere a Möbius transformation?

Just based on some reading, I know that every Möbius transformation is a bijection from the Riemann sphere to itself. I'm curious about the converse. For any holomorphic bijection on the sphere, why ...
### Let $f(z)$ be a one-to-one entire function, Show that $f(z)=az+b$.
Let $f(z)$ be a one-to-one entire function, Show that $f(z)=az+b$. My try : Because $f$ is entire it has a taylor series around zero (in particular). $f(z)=\sum^{\infty}_{k=0} a_kz^k$ Proof by ...