Let $\mathbb D= \{z\in\mathbb{C}: |z|<1\}$. Let $Hol(\mathbb {D})$ denote the space of holomorphic functions on $\mathbb D$. The Hardy spaces on $\mathbb D$ are defined as follows.
$$H^p=\{f\in Hol(\mathbb {D}):\sup_{r<1}\int_{0}^{2\pi} |f(re^{i\theta}|^pd\theta<\infty\}\;\;\;(0<p<\infty)$$
$$H^\infty=\{f\in Hol(\mathbb {D}):\sup_{z\in\mathbb D}|f(z)|<\infty\}$$ To show that $H^q$ is a strict subset of $H^p$ if $0<p<q\leq\infty$, here, they say as a hint to look at functions of the form $f_a(z)=1/(1-z)^a$ for $a>0$. Can anyone explain how this works?
Hint: One idea is to consider the functions $(1-z)^{-a},a>0.$ So we're trying to understand
$$\int_{-\pi}^{\pi}\frac{dt}{|1-re^{it}|^a}$$
as $r\to 1^-.$ To get an idea of what's happening, suppose we brashly let $r=1.$ We're then looking at
$$\int_{-\pi}^{\pi}\frac{dt}{|1-e^{it}|^a}.$$
Now $|1-e^{it}|$ lies between constant mulitples of $|t|,$ so the values of $a$ for which the last integral converges are evident.