Proof: Let $f$ be a Nonconstant Entire Function on the Unit Disc

In summary: Therefore, $|f(z)|\leq r$ for all $|z|<r$, so $f$ maps the open unit disc into itself.In summary, we can prove that $f$, a nonconstant entire function that maps the unit circle, maps the open unit disc into itself by using the maximum modulus principle and the fact that the map $M(r):=\sup_{|z|=r}|f(z)|$ is strictly increasing. This shows that $f$ maps the open unit disc, $\{z: |z| < 1\}$, into itself.
  • #1
Dustinsfl
2,281
5
Let $f$ be a nonconstant entire function that maps the unit circle, $\{z: |z| = 1\}$, into itself. Prove that $f$ maps the open unit disc, $\{z: |z| < 1\}$, into itself.

I am having a little trouble starting this one. z in C
 
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  • #2
Did you try by contradiction, using the maximum modulus principle?
 
  • #3
girdav said:
Did you try by contradiction, using the maximum modulus principle?

How would that be used? An open disc doesn't have a maximum modulus.
 
  • #4
Use the fact that the maximum of the modulus is reached at the boundary.
 
  • #5
girdav said:
Use the fact that the maximum of the modulus is reached at the boundary.

That still doesn't make sense. Every time we get close to the boundary, we can get a little bit closer. Moreover, we can get a little bit closer and infinite amount of times.
 
  • #6
In fact, we have to work with the map $M(r):=\sup_{|z|=r}|f(z)|$. We can show thanks to maximum modulus principle that this map is strictly increasing.
 
  • #7
girdav said:
In fact, we have to work with the map $M(r):=\sup_{|z|=r}|f(z)|$. We can show thanks to maximum modulus principle that this map is strictly increasing.

I don't understand what you are getting at.
 
  • #8
If $M(r_1)\geq M(r_2)$ for some $r_1<r_2$, the maximum modulus principle shows that $f$ is constant, so $M$ is a strcily increasing map. Now, we have that $M(1)=1$, so if $r<1$ then $M(r)<1$.
 

FAQ: Proof: Let $f$ be a Nonconstant Entire Function on the Unit Disc

What is an entire function?

An entire function is a complex-valued function that is defined and holomorphic (differentiable) at every point in the complex plane.

What does it mean for a function to be nonconstant?

A nonconstant function is one that takes on different values at different points, meaning it is not a constant value for all inputs.

What is the unit disc in complex analysis?

The unit disc is the set of complex numbers with a modulus (absolute value) less than or equal to 1, represented as D(0,1) in the complex plane.

What is the significance of proving a theorem about an entire function on the unit disc?

Proving a theorem about an entire function on the unit disc allows us to make conclusions about the behavior of the function on the entire complex plane, as the unit disc is the largest open subset of the complex plane where the function is defined and holomorphic.

Can you provide an example of a nonconstant entire function on the unit disc?

One example of a nonconstant entire function on the unit disc is the function f(z) = z^2 - 1. This function has zeros at z = 1 and z = -1, but takes on different values for other inputs, making it nonconstant.

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