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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[itex]_{k}[/itex]) = 0 for 1 ≤ k ≤ n, show that
|f(z)| ≤ M [itex]\prod |z - z_{k}|/|1 - \bar{z_{k}}z|[/itex] 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!
Let f be holomorphic in D = {z: |z| < 1} and suppose that |f(z)| ≤ M for all z in D.
If f(z[itex]_{k}[/itex]) = 0 for 1 ≤ k ≤ n, show that
|f(z)| ≤ M [itex]\prod |z - z_{k}|/|1 - \bar{z_{k}}z|[/itex] 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!