Suppose f is non-vanishing holomorphic function in $\mathbb D$ and if $|z|=1\to |f(z)|=1$ then we have to show that f is constant
I had learned to extend function using Schwartz reflection Priciple to lower half plane with the condition given
As per Hint Provided, I have to extend f to whole $\mathbb C$ by $f(z)=1/ \overline{{f(\bar{z})}}$ whenever $|z|>1$
I do not know How to proceed?
Any Help will be appreciated
Best Answer
Since $f$ is holomorphic and non-vanishing on $\mathbb{D}$, $1/f$ is also holomorphic on $\mathbb{D}$. By Maximum Modulus Principle applied to $f$ and $1/f$, your condition implies that $$ 1 \leq |f(z)| \leq 1 \Rightarrow |f(z)| = 1 $$ for every $z\in \mathbb{D}$. Then $f$ achieves maximum modulus in the interior of its domain which implies that it is constant, again by Maximum Modulus Principle.
Another way to approach this is to use Schwarz Lemma, not the reflection principle.