A class of gap series with small growth in the unit disc

Letβ>0and letαbe an integer which is at least2. Iffis an analytic function in the unit discDwhich has power series representationf(z)=∑k=0∞ak zkα,limsupk→∞ (log+|ak|/logk)=α(1+β), then the first author has proved thatfis unbounded in every sector{z∈D:Φ−ϵ<argz<Φ+ϵ, forϵ>0}. A natural conjecture concerning these functions is thatlimsupr→1−(logL(r)/logM(r))>0, whereL(r)is the minimum of|f(z)|on|z|=randM(r)is the maximum of|f(z)|on|z|=r. In this paper, investigations concerning this conjecture are discussed. For example, we prove thatlimsupr→1−(logL(r)/logM(r))=1andlimsupr→1−(L(r)/M(r))=0whenak=kα(1+β).