Sequences of Exponents in Constructing Strongly Annular Products

This paper gives conditions on the behavior of a sequence of holomorphic functions {f _k (z)} and a strictly increasing sequence of positive integers {m _k } that assures the infinite product Pf_k(z^m_k){\Pi f_k(z^{m_k})} is strongly annular. A constructive proof is given that shows if the sequence {f _k (z)} exhibits certain boundary behavior along with a uniform boundedness condition then a number p > 1 exists such that if {m _k } satisfies m _k+1/m _k ≥ p then the above product is strongly annular.