Estimations of norms of powers of functions in certain Banach spaces

Asymptotic estimates of norms of powers of analytic functions in certain Banach spaces are obtained. For a function $\varphi$ analytic in the closed unit disc and such that $\sup|\varphi(z)|=1$, it is shown that there exist constants $C,c$ and $\alpha$ depending on $\varphi$ and the Banach space $X$ such that for every $n$
$$ cn^\alpha\le\|\varphi^n\|_X\le Cn^\alpha. $$
The cases in which $X$ is the space $l^p_A$ or the Besov space are considered. Bibliography: 4 titles.