On the singular factor of a linear combination of holomorphic functions

We prove that the linear combinations of functions $f_0,\mathrm{.}\mathrm{.}\mathrm{.},f_n$ in $H^{\mathrm{\infty }}$ have “few” singular inner factors, provided that the $f_j$’s are suitably smooth up to the boundary, while in general this is no longer true.