Boundary Gauss--Lucas type theorems on the disk

TL;DR

This paper extends Gauss--Lucas type theorems to the setting of inner functions on the disk, analyzing the relationship between boundary singularities of functions and their derivatives.

Contribution

It introduces boundary Gauss--Lucas theorems for inner functions and explores conditions linking boundary singularities of functions and their derivatives.

Findings

Inner factor of the derivative is nontrivial under certain conditions.

Boundary singularities of the inner factor influence those of the derivative.

Examples show the conditions are close to optimal.

Abstract

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 $f\in H^\infty$ 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 $f'$ 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.