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

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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.

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Literature Cited

  1. G. Halasz, “On Taylor series absolutely convergent on the circumference of the circle of convergence. III,” Preprint No. 21/1972, Budapest.

  2. D. J. Newman. “Homomorphisms ofl t,”Amer. J. Math.,91, No. 1, 37–46 (1969).

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  3. W. K. Hayman and P. B. Kennedy,Subharmonic Functions, Academic Press, London-New York-San Francisco (1976).

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Authors
  1. M. Y. Blyudze
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  2. S. M. Shimorin
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Additional information

Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 206, 1993, pp. 15–32.

Translated by S. M. Shimorin.

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Blyudze, M.Y., Shimorin, S.M. Estimates of the norms of powers of functions in certain Banach spaces. J Math Sci 80, 1880–1891 (1996). https://doi.org/10.1007/BF02367002

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  • DOI: https://doi.org/10.1007/BF02367002

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