I am going through Rudin's RCA (3rd Ed), and it seems to me that Chapter 17 ($H^p$-Spaces), Exercise 3 (the initial part) has an incorrect equivalency statement. Namely, it states the following:
Suppose $0<p\le \infty$ and $f\in H(U)$. Prove that $f\in H^p$ if and only if there is a harmonic function $u$ in $U$ such that $|f(z)|^p \le u(z)$ for all $z\in U$.
Above $U$ is the unit disk.
I have proved the statement for $0<p<\infty$, but the condition does not hold for $p=\infty$ via a fairly trivial counterexample:
If we take $f(z) = C$, where $C>1$, then clearly $f\in H^{\infty}$ but we also have that $|f(z)|^{\infty} = \infty$.
Is this correct, or I am blindly missing something? I have also skimmed some other texts on Hardy spaces for any relation, but am drawing a blank.
Thanks