For me, the Corona Theorem on the Hardy space states that for all $f_1,...,f_n \in H^{\infty}(\mathbb{D})$ such that $$ \inf_{z \in \mathbb{D}} \sum_{i=1}^n \vert f_i(z) \vert > 0 ,$$ there are $g_1,...,g_n \in H^{\infty}(\mathbb{D})$ such that $\sum_{i=1}^n f_i g_i = 1$. In contrast, the Corona theorem on the Hardy space with bounds states that for every $n \in \mathbb{N}$ and every $\delta > 0$, there is a Constant $ C = C(n,\delta) > 0$ such that for all $f_1,...,f_n \in H^{\infty}(\mathbb{D})$ such that $\Vert f_i\Vert_{\mathbb{D}} \geq 1 $ for $i=1,...,n$ and $$ \inf_{z \in \mathbb{D}} \sum_{i=1}^n \vert f_i(z) \vert \geq \delta > 0,$$ there are $g_1,...,g_n \in H^{\infty}(\mathbb{D})$ such that $\sum_{i=1}^n f_i g_i = 1$ and $\Vert g_i \Vert_{\mathbb{D}} \leq C(n,\delta)$ for alle $i=1,...,n$.
By normalizing, it's quite straightforward that the Corona Theorem with bounds implies the normal one. Now, on the other hand, I am also familiar with Wolff's proof of the Corona Theorem with bounds. In particular, both are true and thus equivalent.
However, I've heard several times now that the version with bounds can be easily/elementary deduced if the classical one holds (that is, without ignoring the assumption and just imitating Wolff's or Carleson's proof). Can anyone give me a hint how to do this?