Estimates of the norms of powers of functions in certain Banach spaces

Abstract

Asymptotic estimates of the norms of powers of analytic functions in certain Banach spaces are obtained. For a function ϕ analytic in the closed unit disk and satisfyingsup |ϕ(z)|=1, it is shown that there exist constants C, c, and α depending on ϕ and a Banach space X such that, for every n, $$cn^\alpha \leqslant \left\| {\varphi ^n } \right\|x \leqslant Cn^\alpha .$$ Cases in which X is the space lp A or the Besov space are considered. Bibliography: 4 titles.