Thread: Maximum Modulus Theorem

Suppose that is entire and that for all with . Prove that has no zeroes in the disk .

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.