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 be the open unit disk in the complex plane C. Let $f\backslash colon\; D\; \backslash to\; D$ be a holomorphic function with f(0)=0. Then

$|\; f(z)\; |\; \backslash le\; |\; z\; |$

for all $z$ in $D$, and

$|\; f\text{'}(0)\; |\; \backslash le\; 1.$

If the equality

$|\; f(z)\; |=|\; z\; |\backslash ,$

holds for any z≠0, or

$|\; f\text{'}(0)\; |=1,\backslash ,$

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.

To prove the lemma, one applies the maximum modulus principle to the function f(z)/z.

Schwarz-Pick theorem

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A variant of the Schwarz lemma can be stated that is invariant under analytic automorphisms on the unit disk, i.e. bijective holomorphic mappings of the unit disc to itself. This variant is known as the Schwarz-Pick theorem:

Let $f\backslash colon\; D\; \backslash to\; D$ be holomorphic. Then, for all $z\_1,z\_2\backslash in\; D,$

$\backslash left|\backslash frac\{f(z\_1)-f(z\_2)\}\{1-\backslash overline\{f(z\_1)\}f(z\_2)\}\backslash right|$

\le \frac{\left|z_1-z_2\right|}{\left|1-\overline{z_1}z_2\right|}

and, for all $z\backslash in\; D$

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

\frac{1}{1-\left|z\right|^2}.

The expression

$d(z\_1,z\_2)=\backslash tanh^\{-1\}\backslash left(\backslash frac\{\backslash left|z\_1-z\_2\backslash right|\}\{\backslash left|1-\backslash overline\{z\_1\}z\_2\backslash right|\}\backslash right)$

is the distance of the points $z\_1,z\_2$ in the Poincaré metric, i.e. the metric in the Poincaré disc model for hyperbolic geometry in dimension two. The Schwarz-Pick theorem then essentially states that a holomorphic map of the unit disk into itself decreases the distance of points in the Poincaré metric. If equality holds throughout in one of the two inequalities above (which is equivalent to saying that the holomorphic map preserves the distance in the Poincaré metric) , then f must be an analytic automorphism of the unit disc, given by a Möbius transformation mapping the unit disc to itself.

An analogous statement on the upper half-plane $\backslash mathbb\{H\}$ can be made as follows:

Let $f\backslash colon\backslash mathbb\{H\}\backslash to\backslash mathbb\{H\}$ be holomorphic. Then, for all $z\_1,z\_2\backslash in\; \backslash mathbb\{H\},$

$\backslash left|\backslash frac\{f(z\_1)-f(z\_2)\}\{\backslash overline\{f(z\_1)\}-f(z\_2)\}\backslash right|$

\le \frac{\left|z_1-z_2\right|}{\left|\overline{z_1}-z_2\right|}.

This is an easy consequence of the Schwarz-Pick theorem mentioned above: One just needs to remember that the Cayley transform $W(z)\; =\; (z-i)/(z+i)$ maps the upper half-plane $\backslash mathbb\{H\}$ conformally onto the unit disc $D$. Then, the map $W\backslash circ\; f\backslash circ\; W^\{-1\}$ is a holomorphic map from $D$ onto $D$. Using the Schwarz-Pick theorem on this map, and finally simplifying the results by using the formula for $W$, we get the desired result. Also, for all $z\backslash in\backslash mathbb\{H\},$

$\backslash frac\{\backslash left|f\text{'}(z)\backslash right|\}\{\backslash mbox\{Im\; \}f(z)\}\; \backslash le\; \backslash frac\{1\}\{\backslash mbox\{Im\; \}(z)\}.$

If equality holds for either the one or the other expressions, then f must be a Möbius transformation with real coefficients. That is, if equality holds, then

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

with $a,b,c,d$ being real numbers, and $ad-bc>0$.

Further generalizations

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The Schwarz-Ahlfors-Pick theorem provides an analogous theorem for hyperbolic manifolds.

De Branges' theorem, formerly known as the Bieberbach Conjecture, is an important extension of the lemma, giving restrictions on the higher derivatives of f at 0 in case f is injective.

References

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

Complex analysis | Hyperbolic geometry | Riemann surfaces | Mathematical theorems

施瓦茨引理