A question on Bloch functions

Question

Let $\mathcal{B}(\Delta)$ be the space of Bloch functions in the unit disk $\Delta$. For any $f\in \mathcal{B}(\Delta)$, we define the Bloch norm by $$ \|f\|_{\mathcal{B}}=\sup_{|z|<1}|f'(z)|(1-|z|^2)+|f(0)|. $$

It seems that some had used the following "facts" $$ X_{1/2}:=\left\{f\in \mathcal{B}(\Delta): \|f\|_{\mathcal{B}}=\sup_{|z|\leq \frac{1}{2}}|f'(z)|(1-|z|^2)+|f(0)| \right\} $$ contains no interior point. While, I forgot the exact statements and the reference.

Any comments and reference will be appreciated.

Answer 1

As stated this property cannot be true. Consider $f(z)=z$. Clearly $f \in X_\frac12$. Let any other $g\in \mathcal{B}$ such that $\Vert g \Vert_\mathcal{B} < \varepsilon$. Then we have that $|f'(0)+g'(0)| > 1-\varepsilon $ and for $|z|\geq 1/2.$ $$ |f'(z)+g'(z)|(1-|z|^2) \leq \frac34 +\varepsilon $$ So if $\varepsilon < \frac38$ then necessarily $f+g\in X_\frac12$.