TITLE: a problem on nonconstant holomorphic function has a zero or no in the closed unit disk
QUESTION [5 upvotes]: Let $f:D \to \mathbb{C}$ be a non-constant holomorphic function ($D$ is the closed unit disk) such that $|f(z)|=1$ for all $z$ satisfying $|z|=1$ . Then prove that there exist $z_0 \in D$ such that $f(z_0)=0$

My thought:-
By Maximum Modulus Theorem $|f(z)|$ has Maximum value on the curve which is $1$. 
By Minimum Modulus Theorem if $f(z)\ne0$ for all $z\in D$, then it has its minimum value on the boundary which is $1$. 
Then $|f(z)|=1$ for all $z\in D$. 
Hence $|f(z)|$ is constant which is a contradiction.   
Is my thinking correct?

REPLY [2 votes]: By Maximum Modulus Theorem $|f(z)|$ has Maximum value on the curve which is $1$ . by Minimum Modulus Theorem if $f(z)\ne0$ for all $z\in D$ Then it has its minimum value on the boundary which is $1$.Then $|f(z)|=1$ for all $z\in D$.Then $|f(z)|$ is constant. Hence $f(z)$ becomes constant which gives a contradiction.