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arXiv · 2607.23399

On the growth of Bloch functions

Bingyang Hu·Jie Xiao·Xiaojing Zhou
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Abstract

We prove that there exist two Bloch functions $f_1$ and $f_2$ on $\mathbb D$ such that $$ |f_1(z)|+|f_2(z)| \geq \left(\log\frac{1}{1-|z|}\right)^{1/2}, \qquad z\in\mathbb D, $$ thereby resolving an open problem posed in 2008 by Girela, Pel\'aez, P\'erez-Gonz\'alez and R\"atty\"a. Our proof is based on a new Szeg\H{o}-type recursion involving $\mathbb C^2$-valued polynomials and their reciprocal polynomials.

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Bingyang Hu, Jie Xiao, Xiaojing Zhou. 2026-07-26. On the growth of Bloch functions. https://arxiv.org/abs/2607.23399

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