Linked Questions

7
votes
1answer
2k views

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!
1
vote
1answer
1k views

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!
2
votes
1answer
794 views

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 ...
0
votes
1answer
686 views

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?
12
votes
2answers
1k views

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.
4
votes
1answer
2k views

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 ...
7
votes
2answers
395 views

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 ...
4
votes
1answer
2k views

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 ...
5
votes
2answers
2k views

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 ...
0
votes
1answer
927 views

transcendental entire function, $Aut(\mathbb{C})$

Well, I was trying to prove $Aut(\mathbb{C})=\{f\in H(\mathbb{C}): f(z)=az+b\}$ for some constant $a,b$, As $f$ is entire these three possibilities can happen $1$. $f(z)=a_o$(constant) $2$. $f(z)=\...
1
vote
0answers
1k views

Conformal map of the complex plane is linear

Gameline Complex Analysis, P. 265 #8 is like this, Show that every conformal self-map of the complex plane $ \mathbb C$ is linear. Hint: The isolated singularity of $f(z)$ must be the simple pole. ...
1
vote
2answers
395 views

$n$-to-$1$ near zero of holomorphic function

Can someone explain to me why a holomorphic function that grows like a polynomial of degree $n$ is $n$-to-$1$ near it's roots? I keep reading this fact on this site, but I can't find an explanation.
2
votes
3answers
98 views

Let $f(z)$ be an entire function with an entire inverse. Prove that as $z$ goes to infinity, $|f(z)|$ goes to infinity.

Prove that $\lim\limits_{z \to \infty} |f(z)| = \infty$ where $f(z)$ is entire and has entire inverse $g(z)$. I can show that the limit cannot be finite since if it were, then we can use Liouville's ...
1
vote
1answer
282 views

Is an analytic one-to-one function on the whole plane necessarily a polynomial? (Can it be disproved?)

I had to show what a one-to-one analytic function from the plane to itself could possibly be. So, I studied the behavior of such a function at infinity: Case 1: Such a function cannot have no ...
0
votes
1answer
31 views

Biholomorphic map $f: \Bbb C \rightarrow \Bbb C$ with $f(z_1) = z_2$ [closed]

Let $z_1, z_2 \in \Bbb C$. I want to find a biholomorphic map $f: \Bbb C \rightarrow \Bbb C$ such that $f(z_1) = z_2$, but I do not know how such a map could look like.

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