The Bohr Radius of the Weighted Bloch Spaces

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 $1\mathrm{/}\sqrt{2}$. A criterion for the sharpness of the Bohr inequality in the weighted Bloch space with $R=1\mathrm{/}\sqrt{2}$ is obtained. Using this criterion, the examples of the weights for which the inequality is sharp is given.