I saw this remark in Hoffman's book - "Banach space of analytic function".
If $f$ is analytic in a neighborhood of $\bar{\mathbb{D}}$, the closure of $\mathbb{D}$; then in the inner-outer factorization $f$, it has has no singular inner part. I am struggling to prove this. Any kind of suggestion is welcome.
what I have found that, if $f$ is analytic analytic in a neighborhood of $\bar{\mathbb{D}}$, and $f$ is a inner function without zero in $\mathbb{D}$, then $f$ must be constant of modulus one. But in general I am not able to show that if I divide the function by its outer part why it still remains analytic in a neighborhood of $\bar{\mathbb{D}}$? If that is true then I am done. Am i going to right direction?
I have another question. If I start with a multiply connected domain $\Omega$ in complex plane instead of unit disk, we know that there is a inner outer factorization (upto modulus) for function in Hardy space $H^2(\Omega)$(Reference: "Function theory in plannar domain" by Fischer). Now again If a function $f$ is analytic in a neighborhood of $\bar{\Omega}$, does it necessarily follow that $f$ has no singular inner part?
With the help of T.A.E I think I got my answer.
As $f$ is holomorphic in a neighborhood $\bar{\mathbb{D}}$, $f$ has finitely many zeros in $\bar{\mathbb{D}}$. Let $a_1,a_2,...,a_n$ be the zeros of $f$ which lies on the circle. So, $f$ can be written as \begin{align} f= B F G \end{align} where, $B$ is the finte Blaschke product, $F = \prod\limits _{i=1}^{n} (z-a_i)$ and $G$ is a holomorphic in a neighbourhood $\bar{\mathbb{D}}$ with no zeros in $\bar{\mathbb{D}}$.
Note that, as $G$ is holomorphic in a neighbourhood of $\bar{\mathbb{D}}$ and has no zeros in $\bar{\mathbb{D}}$, clearly $\log|g|$ is harmonic in a neighborhood of $\bar{\mathbb{D}}$. So $G$ is outer.
Also using the fact, $\int\limits_{0}^{2\pi}\log (|\exp(it)-a|) dt = 0$, where $a$ is a point on the circle, we will have $(z-a)$ is a outer function for $a$ lies on circle. As a consequence $F$ will be outer.
Hence $f$ has no singular inner part.