The Hardy space $H^p$ is defined as the holomorphic functions f on the open unit disk satisfying

$\sup _{0\leq r<1}\big(\frac {1}{2\pi}\int _{0}^{2\pi } |f\left(re^{i\theta }\right)|^{p}\mathrm {d} \theta\big)^{\frac{1}{p}}<\infty$.

I recall reading somewhere that the value in question is increasing in $r$ and the $\sup$ is actually a limit when you take $r$ to approach $1$. If this is true, how would one prove this?

  • $\begingroup$ This requires some knowledge of subharmonic functions. Rudin's book has a proof in the very beginning of the chapter on $H^{p}$ spaces. $\endgroup$ – Kavi Rama Murthy Feb 5 at 23:21

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