The Hardy class of a function with slowly-growing area

Abstract

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