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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Mean growth and smoothness of analytic functions


Author: A. Matheson
Journal: Proc. Amer. Math. Soc. 85 (1982), 219-224
MSC: Primary 30D99; Secondary 46J15
DOI: https://doi.org/10.1090/S0002-9939-1982-0652446-3
MathSciNet review: 652446
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Abstract: Let $ {G_\alpha }$ denote the class of functions $ f(z)$ analytic in the unit disk such that

$\displaystyle \int_0^1 {{{(1 - r)}^{ - \alpha }}{M_\infty }(f',r)} dr < \infty ,$

for some $ \alpha (0 < \alpha < 1)$. A characterization of $ {G_\alpha }$ is given in terms of moduli of continuity and an application is given to Riesz factorization of functions in $ {G_\alpha }$.

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

  • [1] V. P. Havin, The factorization of analytic functions that are smooth up to the boundary, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 22 (1971), 202–205 (Russian). MR 0289783
  • [2] F. A. Šamojan, Division by an inner function in certain spaces of functions that are analytic in the disc, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 22 (1971), 206–208 (Russian). MR 0289786
  • [3] A. Zygmund, Trigonometric series: Vols. I, II, Second edition, reprinted with corrections and some additions, Cambridge University Press, London-New York, 1968. MR 0236587

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DOI: https://doi.org/10.1090/S0002-9939-1982-0652446-3
Article copyright: © Copyright 1982 American Mathematical Society