One-component inner functions II

We continue our study of the set $\mathfrak I_c$ of inner functions $u$ in $H^\infty$ with the property that there is $\eta\in ]0,1[$ such that the level set $\Omega_u(\eta):=\{z\in\mathbb D: |u(z)|<\eta\}$ is connected. These functions are called one-component inner functions. They play an important role in function theory and operator theory. Here we show that the composition of two one-component inner functions is again in $\mathfrak I_c$. We also give conditions under which a factor of one-component inner function belongs to $\mathfrak I_c$.