Let $\mathbb D= \{z\in\mathbb{C}: |z|<1\}$. Let $\operatorname{Hol}(\mathbb {D})$ denote the space of holomorphic functions on $\mathbb D$. The Hardy spaces on $\mathbb D$ are defined as follows.
$$H^p=\{f\in \operatorname{Hol}(\mathbb {D}):\sup_{r<1}\int_{0}^{2\pi} |f(re^{i\theta})|^p d\theta<\infty\}\;\;\;(0<p<\infty)$$
$$H^\infty=\{f\in \operatorname{Hol}(\mathbb {D}):\sup_{z\in\mathbb D}|f(z)|<\infty\}$$
It is known that the function $\ln(1-z)$ belongs to $H^p$ for $0<p<\infty$ but does not belong to $H^\infty$ . Can anyone tell how?