The Bohr Radius of the Weighted Bloch Spaces

TL;DR

This paper investigates the Bohr radius in weighted Bloch spaces, providing improved estimates, a lower bound of 1/√2, and criteria for the sharpness of the Bohr inequality, with examples of weights where the inequality is optimal.

Contribution

It introduces the concept of the Bohr radius for Banach space pairs, improves existing bounds, and establishes criteria for sharpness in weighted Bloch spaces.

Findings

Lower bound for Bohr radius is at least 1/√2.

Improved estimate for Bohr radius from Bloch space to bounded functions.

Criteria for the sharpness of the Bohr inequality in weighted Bloch spaces.

Abstract

The concept of the Bohr radius of a pair of Banach spaces is introduced. The lower estimate for the value of the Bohr radius from the Bloch space to the space of bounded functions obtained by I. Kayumov, S. Ponnusamy and N. Shakirov is slightly improved. It is shown that for any weighted Bloch space the Bohr radius is not less than $1/\sqrt{2}$. A criterion for the sharpness of the Bohr inequality in the weighted Bloch space with $R=1 / \sqrt{2}$ is obtained. Using this criterion, the examples of the weights for which the inequality is sharp is given.