The Hardy Class of a Function with Slowly-Growing Area

In this paper we show that if is analytic on the open unit disk and if the area of <!-- MATH $[\{ |z| \leq R\} \cap {\text{image of }}f]$ --> grows sufficiently slowly as a function of , then belongs to the Hardy class for all satisfying <!-- MATH $0 < p < + \infty$ --> <img width="114" height="37" align="MIDDLE" border="0" src="images/img7.gif" alt="$ 0 < p < + \infty $">.