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Schwarz's lemma, complex analysis proof

  • #1

Homework Statement


Let B1 = {z element C : abs(z) < 1}, f be a holomorphic function on B1 with Re f(z) > greater than or equal to 0 and f(0) =1. then show that:

abs(f(z)) less than or equal to [(1+abs(z))/(1-abs(z))]


Homework Equations


Schwarz's Lemma: Suppose that f is analytic in the unit disc, that abs(f) less than or equal to 1 and that f(0) = 0. Then

i. abs(f(z)) less than or equal to abs(z)
ii. abs(f'(0)) less or equal to 1


The Attempt at a Solution


So I know that the solution to this problem involves utilizing Schwarz's lemma (a hint from my professor), however considering the different value of the point at z = 0 is throwing me for a loop. I am not quite sure how to continue from where I am.
 

Answers and Replies

  • #2
392
0
You have f from B1 to H, where H is the half plane {z : Re(z)>0}.

Now create a function g which maps H into the unit disc, such that g(1)=0.

Let h = composition of g and f. Apply Schwarz to h. After that, a little extra algebra is needed to get the desired conclusion.
 

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