Inverse of a analytic function

Question

Let $f$ be a map on the closed unit disc $\bar{\mathbb{D}}$ in $\mathbb{C}$ such that $f$ is analytic on $\mathbb{D}$ and continuous on $ \bar{\mathbb{D}}$. Can you tell under what condition will $f$ be invertible under point-wise multiplication. Will the inverse also be analytic on $\mathbb{D}$ and continuous on $ \bar{\mathbb{D}}$? If yes why?