Integral means of derivatives of univalent functions in Hardy spaces

Abstract

We show that the norm in the Hardy space H p H^p satisfies ( † ) ‖ f ‖ H p p ≍ ∫ 0 1 M q p ( r , f ′ ) ( 1 − r ) p ( 1 − 1 q ) d r + | f ( 0 ) | p \begin{equation} \|f\|_{H^p}^p\asymp \int _0^1M_q^p(r,f’)(1-r)^{p\left (1-\frac 1q\right )}\,dr+|f(0)|^p\tag {$\dagger $} \end{equation} for all univalent functions provided that either q ≥ 2 q\ge 2 or 2 p 2 + p > q > 2 \frac {2p}{2+p}>q>2 . This asymptotic was previously known in the cases 0 > p ≤ q > ∞ 0>p\le q>\infty and p 1 + p > q > p > 2 + 2 157 \frac {p}{1+p}>q>p>2+\frac {2}{157} by results due to Pommerenke [Math. Ann. 145 (1961/62), pp. 285–296], Baernstein, Girela and Peláez [Illinois J. Math. 48 (2004), pp. 837–859] and González and Peláez [J. Geom. Anal. 19 (2009), pp. 755–771]. It is also shown that ( † ) (\dagger ) is satisfied for all close-to-convex functions if 1 ≤ q > ∞ 1\le q>\infty . A counterpart of ( † ) (\dagger ) in the setting of weighted Bergman spaces is also briefly discussed.