Integral means of derivatives of univalent functions in Hardy spaces

TL;DR

This paper establishes a new integral characterization of the Hardy space norm for univalent functions, extending previous results and including close-to-convex functions, with implications for weighted Bergman spaces.

Contribution

It provides a novel integral formula for Hardy space norms of univalent functions under broader conditions, including close-to-convex functions, and discusses related weighted Bergman space results.

Findings

The norm in Hardy spaces can be characterized by an integral involving derivatives of univalent functions.

The integral characterization holds for all univalent functions when q ≥ 2 or (2p)/(2+p) < q < 2.

The formula also applies to close-to-convex functions for all q ≥ 1.

Abstract

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.