analytic, fixed point

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February 19th 2010, 02:33 PM
eskimo343

analytic, fixed point

Let $f(z)$ be an analytic function mapping the unit disc $\mathbb{D}$ into itself. Suppose that $f(z)$ has two different fixed points in the unit disc $\mathbb{D}$ , show that $f$ must be the identity function $f(z)=z$ for all $z \in \mathbb{D}$ .

$z_0$ is called a fixed point for a function $f$ , if $f(z_0)=z_0$ .

For this one, I was thinking to use Rouché's Theorem or Schwarz lemma. However, we are dealing with fixed points here. So, I don't see what to do right now. I need a few hints on this one. Thanks.
February 19th 2010, 11:46 PM
chisigma

If $f(*)$ is analytic inside the unit disk, then it can be developped in Taylor expansion in [for example...] $z_{0}=0$ ...

$f(z)= \sum_{n=0}^{\infty} \frac{f^{(n)} (0)}{n!}\cdot z^{n}$ (1)

... where the $f^{n} (0)$ are given by the Cauchy integrals...

$f^{n} (0) = \frac{n!}{2 \pi i} \int_{\gamma} \frac{f(z)}{z^{n+1}}\cdot dz$ (1)

... where $\gamma$ is the unit circle. You can verify that the line integral in (2) vanishes forall n with the only exception of n=1 where is is equal to $2\pi i$ . So...

Kind regards

$\chi$ $\sigma$
February 20th 2010, 02:49 AM
Opalg

Alternatively, let the fixed points be $z_1$ and $z_2$ , and let g be a conformal map from the disk to itself that takes $z_1$ to the origin. Then $g f g^{-1}$ is an analytic map from the disk to itself that fixes the origin and also the point $g(z_2)$ . You should now be able to use Schwarz's lemma to show that $g f g^{-1}$ is the identity and hence so is f.
February 20th 2010, 02:57 AM
Laurent

Quote:

Originally Posted by chisigma

You can verify that the line integral in (2) vanishes forall n with the only exception of n=1 where is is equal to $2\pi i$ .

I may be missing something obvious, but I can't see how to prove that. How do you use the two fixed point to compute the integral?
February 20th 2010, 06:07 AM
bkarpuz

Quote:

Originally Posted by Laurent

Quote:

Originally Posted by chisigma

You can verify that the line integral in (2) vanishes forall n with the only exception of n=1 where is is equal to $2\pi i$ .

I may be missing something obvious, but I can't see how to prove that. How do you use the two fixed point to compute the integral?

Actually $\chi\sigma$ proves that every analytic function in the unit disc is constant. Nice and simple. (Rofl)
February 20th 2010, 09:12 AM
chisigma

It is evident that I read hurriedly the post of eskimo 343... with a little more carefull reading it is not difficult to 'discover' that there is an evident counterexample: all the complex functions of the form $f(z)= z^{2n+1}$ are analytic and have two fixed point in $z=1$ and $z=-1$ , both on the unit disc... (Headbang) very sorry!...

Kind regards

$\chi$ $\sigma$
February 20th 2010, 09:59 AM
Opalg

Quote:

Originally Posted by chisigma

It is evident that I read hurriedly the post of eskimo 343... with a little more carefull reading it is not difficult to 'discover' that there is an evident counterexample: all the complex functions of the form $f(z)= z^{2n+1}$ are analytic and have two fixed point in $z=1$ and $z=-1$ , both on the unit disc... (Headbang) very sorry!...

That is correct if you take $\mathbb{D}$ to be the closed unit disk. But I think that the original result is correct if $\mathbb{D}$ is the open disk.