INTEGRAL MEANS OF DERIVATIVES OF UNIVALENT FUNCTIONS IN HARDY SPACES

We show that the norm in the Hardy space H-p satisfies (dagger) parallel to f parallel to(p)(Hp) (sic) integral(1)(0) M-q(p)(r, f')(1 - r)(p(1-1/q)) dr + vertical bar f(0)vertical bar(p) for all univalent functions provided that either q >= 2 or 2p/2+p < q < 2. This asymptotic was previously known in the cases 0 < p <= q < infinity and p/1+p < q < p < 2 + 2/157 by results due to Pommerenke [Math. Ann. 145 (1961/62), pp. 285-296], Baernstein, Girela and Pelaez [Illinois J. Math. 48 (2004), pp. 837-859] and Gonzalez and Pelaez [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 < infinity. A counterpart of (dagger) in the setting of weighted Bergman spaces is also briefly discussed.