Geodesic
Estimations of norms of powers of functions in certain Banach spaces
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 21, Tome 206 (1993), pp. 15-32

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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. 
@article{ZNSL_1993_206_a1,
     author = {M. Yu. Blyudze and S. M. Shimorin},
     title = {Estimations of norms of powers of functions in certain {Banach} spaces},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {15--32},
     publisher = {mathdoc},
     volume = {206},
     year = {1993},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1993_206_a1/}
}
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M. Yu. Blyudze; S. M. Shimorin. Estimations of norms of powers of functions in certain Banach spaces. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 21, Tome 206 (1993), pp. 15-32. http://geodesic.mathdoc.fr/item/ZNSL_1993_206_a1/