I am hoping the following is true but don't yet know. Thoughts and ideas are appreciated.
Let $k \ge 2$ be an integer. Let $0<\epsilon <1$. Then there is a holomorphic function $g: \mathbb{D} \to \mathbb{C}$ s.t. $$ \inf_{z \in \mathbb{D}} |1+z+z^kg(z)| > 1-\epsilon$$
Note: When $k=2$ the conclusion can be seen to follow using the Riemann Mapping Theorem.