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

The Bohr Radius of the Weighted Bloch Spaces