Zeros of the derivatives of a finite Blaschke product.

Question

Let $B$ be an $n$ degree finite Blaschke product. By considering the level curves of $B$, one can show that $B'$ has $n-1$ critical points in the disk (counting multiplicity). Is anything known about the higher derivatives of $B$? For example, the number of zeros of $B'',B''',\ldots$ in the disk?

Answer 1

Maybe it's of interest, particular after a so long non active term. There is a analogue of Gauss-Lucas theorem for Blaschke-Products $B$: It says, that the critical points lie in the non-euclidean convex hull of the zeros of $B$.

I have read it in "Polynomials Versus Finite Blaschke Products" Tuen Wai Ng, Chiu Yin Tsang. Here is the link: http://link.springer.com/chapter/10.1007/978-1-4614-5341-3_14.