PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: High Entry average rating: No information on entry rating
Schwarz lemma (Theorem)

Let $\Delta = \{z: \size{z} < 1\}$ be the open unit disk in the complex plane $\Complex$ Let $f\colon\Delta\to\Delta$ be a holomorphic function with $f(0)=0$ Then $\size{f(z)} \le \size{z}$ for all $z\in\Delta$ and $\size{f'(0)} \le 1$ If the equality $\size{f(z)}=\size{z}$ holds for any $z\ne 0$ or $\size{f'(0)}=1$ then $f$ is a rotation: $f(z)=az$ with $\size{a}=1$

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 lemma" is owned by Koro. [ full author list (3) | owner history (2) ]
(view preamble | get metadata)

View style:


Attachments:
proof of Schwarz lemma (Proof) by Mathprof
Log in to rate this entry.
(view current ratings)

Cross-references: real functions, Riemann mapping theorem, rotation, equality, holomorphic function, complex plane, open unit disk
There are 4 references to this entry.

This is version 5 of Schwarz lemma, born on 2002-06-05, modified 2006-09-13.
Object id is 3047, canonical name is SchwarzLemma.
Accessed 7782 times total.

Classification:
AMS MSC30C80 (Functions of a complex variable :: Geometric function theory :: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination)

Pending Errata and Addenda
None.
[ View all 3 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)