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A subspace X of the Hardy space H 1 is said to have the f-property if h/I ∈ X whenever h ∈ X and I is an inner function with h/I e H 1 . We let B denote the space of Bloch functions and β 0 the little Bloch space. Anderson proved in 1979 that the space β 0 ∩H ∞ does not have the f-property. However, the question of whether or not B ∩ H p (1 < p < ∞) has the f-property was open. We prove that for every p ∈ [1, oo) the space β ∩ H p does not have the f-property. We also prove that if B is any infinite Blaschke product with positive zeros and G is a Bloch function with |G(z)| → ∞, as z → 1, then the product BG is not a Bloch function.

Multiplication and division by inner functions in the space of Bloch functions