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Let $f$ be holomorphic on the unit disc and continuous on the unit circle. Suppose there is an $M \in \mathbb{R}$ such that $|f(z)| \leq M$ on the unit circle and let $\alpha_1, \alpha_2, ..., \alpha_n$ be zeros of $f$ in the unit disc listed according to multiplicity. Show that $|f(z)| \leq M \frac{|z-\alpha_1| \cdots |z- \alpha_n|}{|1-z \overline{\alpha_1}| \cdots |1-z \overline{\alpha_n}|}$.

Why can't I apply the Maximum Modulus theorem to $f$ directly? Is there something I am missing?

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    The factor $M$ is multiplied with may be less than $1$?2012-08-05
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    I think so, there's no mention of it (in our notes) being larger/smaller than 1. That's one big sticking point for me.2012-08-05
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    For example, when $\alpha_j=0$, we should have $|f(z)|\leq Mz^n$, which is a tighter bound than what we get with maximum modulus principle.2012-08-05

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