Regarding $A(D)$ and $H^\infty(D)$

Question

Let $D=\{z\in\mathbb{C}: |z|<1\}$.

$C(\bar{D})=\{f:\bar D\longrightarrow \mathbb{C}: f \;\text{is continuous on}\; \bar{D}\}$

$A(D)=\{f\in C(\bar{D}): f \;\text{is analytic in} \;D\}$

$H^\infty(D)=\{f:D\longrightarrow \mathbb{C}: f \;\text{is bounded and analytic on}\; {D}\}$

All the spaces above are equipped with the sup norm and multiplication is point wise.

I know that the invertible elements in $C(\bar{D})$ are those functions $f$ which never take the value zero on $\bar{D}$.

Can you tell me what exactly are the invertible elements in $A(D)$ and $H^\infty(D)$ and why ?

Answer 1

If $f \in A(D)$ never vanishes then $\frac 1 f \in A(D)$. So invertible elements of $A(D)$ are precisely those elements that never vanish on $\overline D$. Suppose $f,g \in H^{\infty} (D)$ and $fg=1$. Then $\frac 1 f \equiv g$ is bounded on $D$. The converse is also true. So invertible elements of $H^{\infty} (D)$ are precisely those bounded analytic funtions $f$ for which there exists $\epsilon >0$ such that $|f(z)| > \epsilon$ for all $z \in D$.