On the growth of Bloch functions

Abstract

We prove that there exist two Bloch functions $f_1$ and $f_2$ on $\mathbb{D}$ such that $$|f_1(z)|+|f_2(z)|\ge {\left(\log \frac{1}{1-|z|}\right)}^{1/2},z\in \mathbb{D},$$ thereby resolving an open problem posed in 2008 by Girela, Peláez, Pérez-González and Rättyä. Our proof is based on a new Szegő-type recursion involving ${\mathbb{C}}^2$-valued polynomials and their reciprocal polynomials.

arXiv comment10 pages. Comments welcome!