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We define the notion of a Kirby element of a ribbon category $ \ mathcal { C } $ ( not necessarily semisimple ) . We characterize a class of Kirby elements , the algebraic Kirby elements , in terms of the structure maps of a Hopf algebra in $ \ mathcal { C } $ . We obtain Bohr type inequalities for the algebras of bounded analytic functions and the multiplier algebras of reproducing kernel Hilbert spaces on various higher-dimensional domains , for the non-commutative disc algebra $ { \ mathcal A } _ n $ , and for the reduced ( respectively full ) group C * - algebra of the free group on $ n $ generators . In this case , if the subgroup admits a boundary in the sense of $ \ mathcal { Z } $ - structures , we prove that $ \ Gamma $ admits a boundary .