a problem on nonconstant holomorphic function has a zero or no in the closed unit disk

Question

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 $f(z_0)=0$

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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?

Answer 1

Suppose $f(z)\ne 0\in D\forall z$, $g(z)={1\over f(z)}$ is Holomorphic on $D$ and $|g(z)|=1$ on $|z|=1$, and by MMP $|g(z)|$ attains Maxima $1$ on $|z|=1$ so$|g(z)| \le 1\forall z \in D$ but then $|f(z)|>1 \Leftrightarrow$

Answer 2

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.