another problem for the practice exam. i think this too requires an application of schwarz lemma, but i can't even get started. any help is appreciated! thank you.
Let f: D(0,1)-->C (where D(0,1) is the unit disk and C the complex field) be an analytic function such that |f(z)|<1 for all z in D(0,1).
(a) Next, suppose there exist two distinct points a,b in D(0,1) such that f(a)=a and f(b)=b. Show that f(z)=z for all z in D(0,1).
(b) Finally, suppose there exist a in D(0,1), a not equal to 0, such that f(a)=0=f(-a). Show that
What can we conclude if |f(0)|=|a|^2