Suppose that is entire and that for all with . Prove that has no zeroes in the disk .
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Originally Posted by tarheelborn Suppose that is entire and that for all with . Prove that has no zeroes in the disk . Hint Denote and apply the Rouche's Theorem to .
We haven't had Rouche's Theorem. I have the Maximum Modulus Theorem and I am not really sure how to apply that here.
The maximum modlus theorem states if g is holomorphic, |g| reaches its maximum only on the boundary. Now g=f-1 is holomorphic, and |g|<1 on |z|=1, so |g|<1 in the disk |z|<1. So |f|=|1+g|>=|1-|g||>|1-1|=0 in the disk.
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