  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.
