Hardy spaces of close-to-convex functions and their derivatives

Abstract:

Let $f(z) = \sum \nolimits _1^\infty {{a_n}} {z^n}$ be close-to-convex on the unit disc. It is shown that (a) if $\lambda > 0$, then f belongs to the Hardy space ${H^\lambda }$ if and only if ${\sum {{n^{\lambda - 2}}\left | {{a_n}} \right |} ^\lambda }$ is finite and that (b) if $0 < \lambda < 1$, then $f’ \in {H^\lambda }$ if and only if either $\sum {{n^{2\lambda - 2}}} {\left | {{a_n}} \right |^\lambda }$ or, equivalently, $\int _0^1 {{M^\lambda }(r,f’)} dr$ is convergent. It is noted that the first of these results does not extend to the full class of univalent functions and that the second is best possible in a number of different senses.