The definition of Hardy spaces for the unit disk is here.
It is clear that for $0<p<q\le\infty$, $H^q\subseteq H^p$, by Hölder's inequality. I'm asked to show that the inclusion relation is proper. Specifically,
(1) Show that $H^q\subsetneq H^p$, for $0<p<q\le\infty$.
(2) Show that $H^\infty\subsetneq\bigcap_{0<p<\infty}H^p$.
I failed to construct such holomorphic functions. Can anyone help me?
By the way, is it true that $\bigcup_{0<p<\infty}H^p\subsetneq N$, where $N$ is Nevanlinna space?
Hints: (1) Look at functions of the form $f_a(z)=1/(1-z)^a$ for $a>0.$ Note that on the boundary we have
$$|f_a(e^{it})|^p = |1-e^{it}|^{-ap}$$
and that near $t=0,$ $|1-e^{it}| \approx |t|.$
(2) Consider $\ln (1-z).$
The last question: You could try $f(z) = \exp {(1+z)/(1-z)}.$ This is in the Nevanlinna class, because $\log^+|f(re^{it})|$ has a harmonic majorant (namely the Poisson kernel). So try to show $f\not \in H^p, 0<p<\infty,$