Let $H^p(D)$ denote the Hardy space on the open unit disk in the complex plane with $0
Consider $f\in H^r(D)$ and $g\in H^s(D)$ for $0 Does there exist a $0 Related question: Let $N$ denote the Nevanlinna class, i.e. the class of holomorphic functions on the disc $f$ s.t. $$\sup_{0\le r<1} \int_0^{2\pi} \log^+|f(re^{i\theta})| d\theta<\infty$$
where $\log^+(t)=\max\{0, \log(t)\}$ for $t>0$. Then for $f,g\in N$ do we know that $f\cdot g\in N$? An answer to any of the two is welcome.