Products of unbounded Bloch functions

TL;DR

This paper constructs examples of analytic functions in the unit disc where an unbounded Bloch function multiplied by a bounded analytic function results in a non-Bloch function, revealing new insights into the product behavior of Bloch functions.

Contribution

It introduces novel constructions of function pairs demonstrating that the product of a bounded and an unbounded Bloch function need not be Bloch, challenging previous assumptions.

Findings

Product of bounded and unbounded Bloch functions can fail to be Bloch

New explicit examples of such function pairs

Insights into the structure of Bloch functions and their products

Abstract

We give new constructions of pair of functions $(f, g)$, analytic in the unit disc, with $g\in H^\infty $ and $f$ an unbounded Bloch function, such that the product $g\cdot f$ is not a Bloch function.