schwarz lemma

In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions defined on the open unit disk.

Let

\Delta = \{z: | z | < 1\}

be the open unit disk in the complex plane C. Let

f:\Delta\to\Delta

be a holomorphic function with f(0)=0. Then

| f(z) | \le | z |

for all

z

\Delta

, and

| f'(0) | \le 1

. If the equality

| f(z) |=| z |\,

holds for any z≠0, or

| f'(0) |=1\,

then

f

is a rotation:

f(z)=az

, with

| a |=1

. This lemma is less celebrated than the bigger guns (such as the Riemann mapping theorem, which it helps prove); however, it is one of the simplest results capturing the "rigidity" of holomorphic functions. No similar result exists for real functions, of course.

Schwarz-Pick theorem

A variant of the Schwarz lemma can be stated that is invariant under a change of coordinates on the unit disk. This variant is known as the Schwarz-Pick theorem: Let

f:\Delta\to\Delta

be holomorphic. Then, for all

z_1,z_2\in \Delta

\left|\frac{f(z_1)-f(z_2)}{1-\overline{f(z_1)}f(z_2)}\right|

\le \frac{\left|z_1-z_2\right|}{\left|1-\overline{z_1}z_2\right|} and, for all

z\in\Delta

\frac{\left|f'(z)\right|}{1-\left|f(z)\right|^2} \le

\frac{1}{1-\left|z\right|^2}. If equality holds for either the one or the other expression, then f must be a Mbius transformation, in which case both expressions are identities. An analogous statement on the upper half-plane

\mathbb{H}

can be made as follows: Let

f:\mathbb{H}\to\mathbb{H}

be holomorphic. Then, for all

z_1,z_2\in \mathbb{H}

\left|\frac{f(z_1)-f(z_2)}{\overline{f(z_1)}-f(z_2)}\right|

\le \frac{\left|z_1-z_2\right|}{\left|\overline{z_1}-z_2\right|} and, for all

z\in\mathbb{H}

\frac{\left|f'(z)\right|}{\mbox{Im }f(z)} \le

\frac{1}{\mbox{Im }(z)}. If equality holds for either the one or the other expression, then f must be a Mbius transformation with real coefficients, in which case both expressions are identities. That is, if equality holds, then

f(z)=\frac{az+b}{cz+d}

with

a,b,c,d

being real numbers, and

ad-bc>0

Further generalizations

The Schwarz-Ahlfors-Pick theorem provides an analogous theorem for hyperbolic manifolds. Louis De Branges' theorem is an important generalization.

References

Jurgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York. ISBN 3-540-43299-X (See Section 2.3)

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