The question comes from the answer to this question Finding maximum of this function in complex analysis
The answer claims that "hence $g/B$ is a holomorphic function from the unit disc to itself".
We only know that $g: D \to D$ and $B: D \to D$ and $g/B$ is holomorphic with removable singularities, right? Why does that guarantee $g/B: D \to D$?
$|\frac g B| \leq1$ in $|z|=1$. By Maximum Modulus Principle the inequality holds for $|z| <1$ also. By Open Mapping Theorem for analytic functions the image of the open unit disk under $\frac g B$ is an open set so this image must be contained in $D$.