Boundary Gauss--Lucas type theorems on the disk

The classical Gauss–Lucas theorem describes the location of the critical points of a polynomial. There is also a hyperbolic version, due to Walsh, in which the role of polynomials is played by finite Blaschke products on the unit disk. We consider similar phenomena for generic inner functions, as well as for certain “locally inner” self-maps of the disk. More precisely, we look at a unit-norm function $f\in H^{\mathrm{\infty }}$ that has an angular derivative on a set of positive measure (on the boundary) and we assume that its inner factor, $I$, is nontrivial. Under certain conditions to be discussed, it follows that $f^{\mathrm{\prime }}$ must also have a nontrivial inner factor, say $J$, and we study the relationship between the boundary singularities of $I$ and $J$. Examples are furnished to show that our sufficient conditions cannot be substantially relaxed.