Integral means of derivatives of univalent functions in Hardy spaces

We show that the norm in the Hardy space $H^p$ satisfies \begin{equation}\label{absteq} \|f\|_{H^p}^p\asymp\int_0^1M_q^p(r,f')(1-r)^{p\left(1-\frac1q\right)}\,dr+|f(0)|^p\tag{\dag} \end{equation} for all univalent functions provided that either $q\ge2$ or $\frac{2p}{2+p}<q<2$. This asymptotic was previously known in the cases $0<p\le q<\infty$ and $\frac{p}{1+p}<q<p<2+\frac{2}{157}$ by results due to Pommerenke (1962), Baernstein, Girela and Pel\'aez (2004) and Gonz\'alez and Pel\'aez (2009). It is also shown that \eqref{absteq} is satisfied for all close-to-convex functions if $1\le q<\infty$. A counterpart of \eqref{absteq} in the setting of weighted Bergman spaces is also briefly discussed.