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Proceedings of the American Mathematical Society

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Mean growth of inner functions

Author: Domingo A. Herrero
Journal: Proc. Amer. Math. Soc. 37 (1973), 175-180
MSC: Primary 30A76
MathSciNet review: 0308403
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Abstract: Let A be a closed subset of the closed unit disc. It is shown that there exists a ``universal growth function'' $ \varphi (r,A)$ such that $ 1 - \smallint _0^{2\pi }\vert q(r{e^{ix}}){\vert^2}dx/2\pi = O(\varphi (r,A))$ for all inner functions $ q(z)$ whose zeroes lie in $ A \cap \{ \vert z\vert < 1\} $ and whose singularities in the unit circle lie on $ A \cap \{ \vert z\vert = 1\} $, if and only if the Lebesgue measure of $ A \cap \{ \vert z\vert = 1\} $ is zero.

References [Enhancements On Off] (What's this?)

  • [1] D. A. Herrero, Inner function-operators, Dissertation, University of Chicago, Chicago, Ill., 1970. MR 0634685 (58:30364)
  • [2] K. Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1962. MR 24 #A2844. MR 0133008 (24:A2844)
  • [3] J. E. Littlewood, Lectures on the theory of functions, Oxford Univ. Press, Oxford, 1944. MR 6, 261. MR 0012121 (6:261f)
  • [4] D. J. Newman and H. S. Shapiro, The Taylor coefficients of inner functions, Michigan Math. J. 9 (1962), 249-255. MR 26 #6371. MR 0148874 (26:6371)

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Keywords: Mean growth in $ {L^2}$-norm, inner function, Blaschke product, singular part, singular measure, singularities in the boundary
Article copyright: © Copyright 1973 American Mathematical Society

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