A Class of Gap Series with Small Growth in the Unit Disc

Let β > 0 and let α be an integer which is at least 2. If f is an analytic function in the unit disc D which has power series representation f(z) =∑∞k=0akzkα , limsupk→∞(log+ |ak|/ logk) = α(1+β), then the first author has proved that f is unbounded in every sector {z ∈D :φ− < argz < φ+ , for > 0}. A natural conjecture concerning these functions is that limsupr→1−(logL(r)/ logM(r)) > 0, where L(r) is theminimumof |f(z)| on |z| = r andM(r) is themaximum of |f(z)| on |z| = r . In this paper, investigations concerning this conjecture are discussed. For example, we prove that limsupr→1−(logL(r)/ logM(r))= 1 and limsupr→1−(L(r)/M(r))= 0 when ak = kα(1+β).