In the theory of Hardy spaces of the unit disc, a fact that is implicitely used quite often is that if $f\in H^p, 1<p<\infty$, then there exists a function $F\in H^p$ such that $|f(z)| \leq |F(z)|, \,\, \forall z \in \mathbb{D}$, $ Re F \geq 0$ and $ \Vert F \Vert_p \leq c_p \Vert f \Vert_p $.

To see why this is the case, given $f\in H^p$ define \begin{equation*} F(z) = \int_{\mathbb{T}} |f(\zeta)| \frac{1+\overline{\zeta}z}{1-\overline{\zeta}z} |d\zeta|. \end{equation*}

This is sometimes called Herglotz transform of $|f|$, but the point is that is a bounded linear operator from $L^p(\mathbb{T})$ into $H^p$, as a corollary of the M. Riesz Theorem. Hence $F$ defined like this has the required properties.

I was wondering if the existence of such an $F$ could be also true in the case $p=1$. Although the construction should be completely different because of the Failure of the M. Riesz Theorem for $p=1$.