A class of meromorphic functions of slow growth in the unit disk not containing any of their integrals

The class F consists of those functions with the property lim supr→1−T(r,f)−log(1−r)=α(f)<+∞. Here we study the subclass S of F which consists of those functions in F with integrals not in F. In a sense the functions in S form a boundary for F in that all of their derivatives are in F, but any number of integrals of functions in S are not in F. We analyze the relationships between S and functions in Hardy, Bergman, and Dirichlet spaces. We also consider the power series representation of functions in S.