Schwarz lemma

Let be the open unit disk in the complex plane $\backslash mathbb\{C\}$ centered at the origin, and let $f\; :\; \backslash mathbf\{D\}\backslash rightarrow\; \backslash mathbb\{C\}$ be a holomorphic map such that $f(0)\; =\; 0$ and $|f(z)|\backslash leq\; 1$ on $\backslash mathbf\{D\}$.

Then $|f(z)|\; \backslash leq\; |z|$ for all $z\; \backslash in\; \backslash mathbf\{D\}$, and $|f\text{'}(0)|\; \backslash leq\; 1$.

Moreover, if $|f(z)|\; =\; |z|$ for some non-zero $z$ or $|f\text{'}(0)|\; =\; 1$, then $f(z)\; =\; az$ for some $a\; \backslash in\; \backslash mathbb\{C\}$ with $|a|\; =\; 1$.Theorem 5.34 in {{cite book|last1=Rodriguez|first1=Jane P. Gilman, Irwin Kra, Rubi E.|title=Complex analysis : in the spirit of Lipman Bers|date=2007|publisher=Springer|location=New York|isbn=978-0-387-74714-9|page=95|edition=[Online]}}

The proof is a straightforward application of the maximum modulus principle on the function

:$g(z)\; =\; \backslash begin\{cases\}$

\frac{f(z)}{z}\, & \mbox{if } z \neq 0 \\

f'(0) & \mbox{if } z = 0,

\end{cases}

which is holomorphic on the whole of $D$, including at the origin (because $f$ is differentiable at the origin and fixes zero). Now if $D\_r\; =\; \backslash \{z\; :\; |z|\; \backslash le\; r\backslash \}$ denotes the closed disk of radius $r$ centered at the origin, then the maximum modulus principle implies that, for , given any $z\; \backslash in\; D\_r$, there exists $z\_r$ on the boundary of $D\_r$ such that

:$|g(z)|\; \backslash le\; |g(z\_r)|\; =\; \backslash frac$

As $r\; \backslash rightarrow\; 1$ we get $|g(z)|\; \backslash leq\; 1$.

Moreover, suppose that $|f(z)|\; =\; |z|$ for some non-zero $z\; \backslash in\; D$, or $|f\text{'}(0)|\; =\; 1$. Then, $|g(z)|\; =\; 1$ at some point of $D$. So by the maximum modulus principle, $g(z)$ is equal to a constant $a$ such that $|a|\; =\; 1$. Therefore, $f(z)\; =\; az$, as desired.

A variant of the Schwarz lemma, known as the Schwarz–Pick theorem (after Georg Pick), characterizes the analytic automorphisms of the unit disc, i.e. bijective holomorphic mappings of the unit disc to itself:

Let $f:\; \backslash mathbf\{D\}\backslash to\backslash mathbf\{D\}$ be holomorphic. Then, for all $z\_1,z\_2\backslash in\backslash mathbf\{D\}$,

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

and, for all $z\backslash in\backslash mathbf\{D\}$,

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

The expression

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

is the distance of the points $z\_1$, $z\_2$ in the Poincaré metric, i.e. the metric in the Poincaré disk 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 $\backslash mathbf\{H\}$ can be made as follows:

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

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

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 $\backslash mathbf\{H\}$ conformally onto the unit disc $\backslash mathbf\{D\}$. Then, the map $W\backslash circ\; f\backslash circ\; W^\{-1\}$ is a holomorphic map from $\backslash mathbf\{D\}$ onto $\backslash mathbf\{D\}$. 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 $z\backslash in\backslash mathbf\{H\}$,

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

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 $a,b,c,d\backslash in\backslash mathbb\{R\}$ and $ad-bc>0$.

The proof of the Schwarz–Pick theorem follows from Schwarz's lemma and the fact that a Möbius transformation of the form

:

maps the unit circle to itself. Fix $z\_1$ and define the Möbius transformations

: $M(z)=\backslash frac\{z\_1-z\}\{1-\backslash overline\{z\_1\}z\},\; \backslash qquad\; \backslash varphi(z)=\backslash frac\{f(z\_1)-z\}\{1-\backslash overline\{f(z\_1)\}z\}.$

Since $M(z\_1)=0$ and the Möbius transformation is invertible, the composition $\backslash varphi(f(M^\{-1\}(z)))$ maps $0$ to $0$ and the unit disk is mapped into itself. Thus we can apply Schwarz's lemma, which is to say

:$\backslash left\; |\backslash varphi\backslash left(f(M^\{-1\}(z))\backslash right)\; \backslash right|=\backslash left|\backslash frac\{f(z\_1)-f(M^\{-1\}(z))\}\{1-\backslash overline\{f(z\_1)\}f(M^\{-1\}(z))\}\backslash right|\; \backslash le\; |z|.$

Now calling $z\_2=M^\{-1\}(z)$ (which will still be in the unit disk) yields the desired conclusion

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

To prove the second part of the theorem, we rearrange the left-hand side into the difference quotient and let $z\_2$ tend to $z\_1$.

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; that is, univalent.

The Koebe 1/4 theorem provides a related estimate in the case that $f$ is univalent.

• Nevanlinna–Pick interpolation

• Jurgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York. {{isbn|3-540-43299-X}} (See Section 2.3)
• {{cite book | author = S. Dineen | title = The Schwarz Lemma | publisher = Oxford | year = 1989 | isbn = 0-19-853571-6 | url-access = registration | url = https://archive.org/details/schwarzlemma0000dine }}

{{PlanetMath attribution|title=Schwarz lemma|id=3047}}

Category:Riemann surfaces

Category:Lemmas in analysis

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