Let $H^{\infty}$ denote the space of bounded analytic functions on the unit disk $\mathbb{D}$. Let $BMOA$ be the subspace of Hardy Space $H^2$, such that $$\|f\|_{BMOA}=|f(0)|+\sup_{a\in\mathbb{D}}\|f\circ \phi_a-f(a)\|_2<+\infty$$ where $$\left\|\sum_{n=0}^{+\infty}a_nz^n\right\|_2^2=\sum_{n=0}^{+\infty}|a_n|^2$$ and $$\phi_a(z)=\frac{a-z}{1-\overline{a}z}\qquad a,z\in\mathbb{D}.$$
It is certainly true that $$\overline{H^{\infty}}^{BMOA}\subset BMOA.$$ Does there exist a function $f\in BMOA$, such that $f\not\in\overline{H^{\infty}}^{BMOA}$?
Kind regards.