I'm working on an exercise from Peter Duren's Theory of $H^p$ Spaces. The question is:
Show that $(1-z)^{-1}$ is in $H^p$ for every $p < 1$, but not in $H^1$.
I have been able to show the result for $H^1$ but for $p<1$, I'm stuck. The definition of $H^p$ space is rather difficult to work with (at least for me).
I would like to show that $\sup\limits_{0 < r < 1}\left(\frac{1}{2\pi}\int_0^{2\pi}\frac{dt}{|1-re^{it}|^p}\right)^{\frac{1}{p}}=:M_p ~<~ \infty.$