Abstract
Bourgain showed in his paper that if $|g(z)|\leq c\alpha (|f_1(z)|+|f_2(z)|+...+|f_n(z)|), \forall z\in D$ for some function with $\lim_{t\to 0}\frac{\alpha (t)}{t}=0$, where $g, f_1, f_2, ..., f_n \in H^\infty (D)$ . Then g belongs to the norm closure of the idea $I(f_1, f_2, ..., f_n)$ . However, he also showed that the theorem fails if $\alpha (t)=t$ . In this note, following the idea in Bourgain's paper, we construct explicitly two Blaschke products $B_1$ and $B_2$ on the unit disc, which obviously satisfy $|B_1B_2|\leq \frac{1}{2}(|B_1^2|+|B_2^2|)$, such that $B_1B_2$ does not belong to the norm closure of $I(f_1, f_2, ..., f_n)$ .