# Application of Schwarz Lemma-Complex Analysis

1. May 8, 2012

### 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!

2. Jun 19, 2012

### DonAntonio

Re: Application of Schwarz Lemma--Complex Analysis

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

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