Application of Schwarz Lemma-Complex Analysis

[I Love Math]

Application of Schwarz Lemma--Complex Analysis

Let f be holomorphic in D = {z: |z| < 1} and suppose that |f(z)| ≤ M for all z in D.

If f(z_{k}) = 0 for 1 ≤ k ≤ n, show that

|f(z)| ≤ M \prod |z - z_{k}|/|1 - \bar{z_{k}}z| for k = 1 to n and |z| < 1.


I am just very confused about how to begin this problem. We know that f is bounded and holomorphic, and I feel like Schwarz's lemma is relevant here, and maybe a Mobius transformation. Any help in getting me started would be much appreciated. Thanks!

 

[DonAntonio]

Let f be holomorphic in D = {z: |z| < 1} and suppose that |f(z)| ≤ M for all z in D.

If f(z_{k}) = 0 for 1 ≤ k ≤ n, show that

|f(z)| ≤ M \prod |z - z_{k}|/|1 - \bar{z_{k}}z| for k = 1 to n and |z| < 1.


I am just very confused about how to begin this problem. We know that f is bounded and holomorphic, and I feel like Schwarz's lemma is relevant here, and maybe a Mobius transformation. Any help in getting me started would be much appreciated. Thanks!

I think you're right, yet I feel that what you want is actually the Schwarz-Pick Lemma, which you can find in number 3 in [1]

DonAntonio

[1] http://en.wikipedia.org/wiki/Schwarz_lemma#Schwarz.E2.80.93Pick_theorem