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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 s.t. $f\cdot g\in H^t(D)$ ?

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.

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For the first question, assume that $s\leq t$; then using Jensen's inequality we get that $f,g\in H^s(D)$. We have $2|f(re^{i\theta})|^{s/2}|g(re^{i\theta})|^{s/2}\leq |f(re^{i\theta})|^s+|g(re^{i\theta})|^s,$ so $f\cdot g\in H^{t/2}(D)$ (the case $t=\infty$ is obvious).

We have $\log^+t=\frac{\log t+|\log t|}2$, so for $a,b>0$ $\log^+(ab)=\frac{\log a+\log b+|\log a+\log b|}2\leq \log^+a+\log^+b,$ which gives the result.

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    Right, fixed now.2012-10-29