Complex analysis - Schwarz's Lemma

In summary: Therefore, we have M|\frac{f(z)-f(z_{0})}{M^2-\overline{f(z_{0})}f(z)}|≤|g(z)-h(z)|+|h(z)|. Now, we can apply the triangle inequality again to get |g(z)-h(z)|+|h(z)|≤|g(z)|+|h(z)|. Hence, we have M|\frac{f(z)-f(z_{0})}{M^2-\overline{f(z_{0})}f(z)}|≤|g(z
  • #1
d2j2003
58
0

Homework Statement



Show that M|[itex]\frac{f(z)-f(z_{0})}{M^2-\overline{f(z_{0})}f(z)}[/itex]|≤|[itex]\frac{z-z_{0}}{1-\overline{z_{0}}z}[/itex]|

for all z, z[itex]_{0}[/itex] in {w:|w|<1|

Homework Equations



I think I will have to use Schwarz's Lemma: if f is analytic in disc |z|<1, f(0)=0, and |f(z)|≤1 for all z in the disc, then |f(z)|≤|z| , |z|<1

The Attempt at a Solution



This looks similar to Schwarz Pick Theorem (found online, don't completely understand..)

but one of the steps they used was to fix z which would mean that f(z)=z so the equations would become M|[itex]\frac{z-f(z_{0})}{M^2-\overline{f(z_{0})}z}[/itex]|≤|[itex]\frac{f(z)-z_{0}}{1-\overline{z_{0}}f(z)}[/itex]| or M|[itex]\frac{z-f(z_{0})}{M^2-\overline{f(z_{0})}z}[/itex]|≤|[itex]\frac{z-z_{0}}{1-\overline{z_{0}}z}[/itex]| not sure how that would help though...
 
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  • #2


First, let's define the function g(z) = \frac{f(z)-f(z_{0})}{M^2-\overline{f(z_{0})}f(z)}. We can see that this function is analytic in the disc |z|<1 since f(z) is analytic and the denominator is never 0 in the given domain. Also, g(z_{0}) = \frac{f(z_{0})-f(z_{0})}{M^2-\overline{f(z_{0})}f(z_{0})} = 0.

Now, we can use the Schwarz Lemma by fixing z as the input and f(z) as the output. This means that we are considering the function h(z) = \frac{z-f(z_{0})}{1-\overline{z_{0}}z}. We can see that h(z) is also analytic in the disc |z|<1 and h(z_{0}) = \frac{z_{0}-f(z_{0})}{1-\overline{z_{0}}z_{0}} = 0.

Now, using the Schwarz Lemma, we have |g(z)|≤|h(z)| for all z in the given domain. Substituting in our definitions for g(z) and h(z), we get M|\frac{f(z)-f(z_{0})}{M^2-\overline{f(z_{0})}f(z)}|≤|\frac{z-f(z_{0})}{1-\overline{z_{0}}z}|.

Finally, using the triangle inequality, we can rewrite the left hand side as M|\frac{f(z)-f(z_{0})}{M^2-\overline{f(z_{0})}f(z)}|≤M|\frac{f(z)-f(z_{0})}{M^2-\overline{f(z_{0})}f(z)}-\frac{z-f(z_{0})}{1-\overline{z_{0}}z}|+M|\frac{z-f(z_{0})}{1-\overline{z_{0}}z}|.

Since we already know that |g(z)|≤|h(z)| for all z in the given domain, the first term on the right hand side can be rewritten as |g(z)-h(z)|. Also, using the definition
 

1. What is Schwarz's Lemma in complex analysis?

Schwarz's Lemma is a fundamental result in complex analysis that states that if a function is holomorphic on the open unit disk and takes values in the same disk, then the function is either a rotation or a dilation of the disk. In other words, it maps the unit disk onto itself, preserving the shape and size of the disk.

2. How is Schwarz's Lemma used in complex analysis?

Schwarz's Lemma is used in complex analysis to prove important results, such as the maximum modulus principle and the open mapping theorem. It is also a key tool in the proof of the Riemann mapping theorem, which states that any simply connected region in the complex plane can be conformally mapped onto the open unit disk.

3. What is the proof of Schwarz's Lemma?

The proof of Schwarz's Lemma involves using the Cauchy-Riemann equations to show that the derivative of the function is bounded by 1 on the unit disk. This, combined with the maximum modulus principle, leads to the conclusion that the function must be a rotation or dilation of the disk.

4. What are some applications of Schwarz's Lemma?

Schwarz's Lemma has many applications in complex analysis, including in the study of conformal mappings, analytic functions, and the theory of holomorphic automorphisms. It is also used in other areas of mathematics, such as in the study of partial differential equations and in geometric function theory.

5. Are there generalizations of Schwarz's Lemma?

Yes, there are several generalizations of Schwarz's Lemma, including the Koebe distortion theorem, which gives an estimate for the distortion of a conformal mapping near a point, and the Bieberbach conjecture, which is a higher-dimensional generalization of Schwarz's Lemma. There are also many variations and extensions of the lemma in different contexts, such as in the theory of infinite-dimensional Banach spaces.

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