Example of a function which is in Hardy space $H^1(\mathbb{D})$ but not in $H^2(\mathbb{D})$

Question

We know that the Hardy spaces on unit disc $\mathbb{D}$, $H^2(\mathbb{D})\subset H^1(\mathbb{D})$. I need to find an example to show that the containment is proper. I was trying to use the fact that product of two $H^2(\mathbb{D})$ functions is contained in $H^1(\mathbb{D})$. I don't know how to approach further.