1. Maximum Modulus Theorem

Suppose that $f$ is entire and that $|f(z)-1|<1$ for all $z$ with $|z|=1$. Prove that $f(z)$ has no zeroes in the disk $\Delta= \{z:|z|<1\}$.

2. Re: Maximum Modulus Theorem

Originally Posted by tarheelborn
Suppose that $f$ is entire and that $|f(z)-1|<1$ for all $z$ with $|z|=1$. Prove that $f(z)$ has no zeroes in the disk $\Delta= \{z:|z|<1\}$.
Hint Denote $g(z)=f(z)-1$ and apply the Rouche's Theorem to $f(z)=g(z)+1$ .

3. Re: Maximum Modulus Theorem

We haven't had Rouche's Theorem. I have the Maximum Modulus Theorem and I am not really sure how to apply that here.

4. Re: Maximum Modulus Theorem

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.