In Duren's book "Theory of $H^p$ spaces" (MSN) in the comment section after Section 4, it is mentioned that Littlewood and Hardy proved in Some properties of conjugate functions that if $u$ is a harmonic function in the unit disc belonging to the harmonic Hardy space $h^p$, $0<p\leq 1$. Then the harmonic conjugate of $u$, call it $v$ satisfies $$ M_p(r,v):= \biggl( \frac{1}{2\pi} \int_{0}^{2\pi} \lvert v(re^{i\theta})\rvert^p d\theta \biggr)^\frac1p= O\bigl( \bigl(\log\frac{1}{1-r}\bigr)^\frac1p \bigr), \,\,\, r\to 1. $$
Furthermore they show that for $p=1,\frac12,\frac13,\dotsc$ the "big-O" cannot be improved to "little-o" by giving a concrete example. Duren then claims that this is an open question for other values of $p\in(0,1]$.
Given that the book is written in the '70s I would like to know if this is still an open problem, or if there has been some progress.
$p$-integral, not "$p-$integral"$p-$integral, for presumably obvious reasons. I have edited accordingly. $\endgroup$