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 {zD:Φϵ<argz<Φ+ϵ, for ϵ>0}. A natural conjecture concerning these functions is that limsupr1(logL(r)/logM(r))>0, 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 limsupr1(logL(r)/logM(r))=1 and limsupr1(L(r)/M(r))=0 when ak=kα(1+β).