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 , limsup k → ∞ (log + | a k |/log k ) = α (1 + β ), then the first author has proved that f is unbounded in every sector { z ∈ D : Φ − ϵ < arg z < Φ + ϵ , for ϵ > 0}. A natural conjecture concerning these functions is that , where L ( r ) is the minimum of | f ( z )| on | z | = r and M ( r ) is the maximum of | f ( z )| on | z | = r . In this paper, investigations concerning this conjecture are discussed. For example, we prove that and when a k = k α (1+ β ) .