Analytic mapping of unit disc onto itself with two fixed pts.

[d2j2003]

Homework Statement

let f(z) be a 1-1 analytic mapping of unit disc |z|<1 onto itself with two fixed points in |z|<1 Show that f(z)=z

Homework Equations

none

The Attempt at a Solution

I'm thinking there has to be a theorem or something that I am missing for this.. But I'm not really sure where to get started..

To start, let a and b be the two fixed points in |z|<1, then f(a)=a and f(b)=b and I am sure there is a way to show that the only function that does this is f(z)=z not sure how though...

 

[Dick]

Try thinking about the Schwarz lemma.

 

[d2j2003]

Got it, Thanks!