[I am New to complex Analysis Just Learned Scwarz reflection principal,I had solved above using that .I wanted to check my approch. Have a look at solution ,If there is gaps in proof if you think , I will happy to see that]
As given $f$ is non vanishing continous function on closed Unit disc and holomorphic on unit disc.
We have to extend function all over c.
We have 3 partition , Inside disc , On boundary and outside disk
We had already definied function on first 2
$|z|>1$ take conjugate to make anticonformal $z\to \bar z$ thentake reciprocal $\bar z\to 1/\bar z$ now $|1/\bar z|<1$ so $f(1/\bar z)$ definied
To make it conformal map again we will take conjugate and reciprocal......[why reciprocal? see downward]
$f:C\to C$ defined as
Now time to use non vanishing assumption that means $f(z)\neq 0 for |z|<1$
by defination [use of reciprocal] for |z|>1 f(z) become bounded.
Now by symmetry principal we have extended funciton to whole $\mathbb C$ and we have bounded function.
By Lioveilie theorem it is constant