The Hardy class of a function with slowly-growing area

Abstract:

In this paper we show that if $f$ is analytic on the open unit disk and if the area of $[\{ |z| \leq R\} \cap {\text {image of }}f]$ grows sufficiently slowly as a function of $R$, then $f$ belongs to the Hardy class ${H^p}$ for all $p$ satisfying $0 < p < + \infty$.