I am reading these notes titled Hardy spaces, inner and outer functions, shift-invariant subspaces, Toeplitz and Hankel operators
In the second last paragraph, it says that any function that is invertible in $H^\infty$ is outer. Can anyone tell how such a function is outer?
If $f\in H^{\infty}$ is invertible in $H^{\infty}$ with inverse $g\in H^{\infty}$, then $$ 1 = |f||g| \implies \ln|f|=-\ln|g| \mbox{ in } \mathbb{D}. $$ Therefore $\ln|f|$ is absolutely integrable because $\ln^{+}|f|$ and $\ln^{-}|f|$ are both integrable.