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Image areas and $ H\sb{2}$ norms of analytic functions


Author: Shōji Kobayashi
Journal: Proc. Amer. Math. Soc. 91 (1984), 257-261
MSC: Primary 30D55; Secondary 30C80
DOI: https://doi.org/10.1090/S0002-9939-1984-0740181-4
MathSciNet review: 740181
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Abstract: For an analytic function $ f$ in the unit disc $ U$ with $ f(0) = 0$, the inequality $ \left\Vert f \right\Vert _2^2 \leqslant \frac{1}{\pi }$ area $ \{ f(U)\} $ is shown, where an equality occurs if and only if $ f$ is a constant multiple of an inner function. As a corollary, it is shown that for an analytic function in a general domain the square of its $ {H_2}$ norm is bounded by its Dirichlet integral, with the equality condition being settled


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

  • [1] H. Alexander, B. A. Taylor and J. L. Ullman, Areas of projections of analytic sets, Invent. Math. 16 (1972), 335-341. MR 0302935 (46:2078)
  • [2] P. L. Duren, Theory of $ {H_p}$ spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
  • [3] O. Frostman, Potentiel d'équilibre et capacité des ensembles avec quelques applications á la théorie des fonctions, Medd. Lunds Univ. Mat. Sem. 3 (1935), 1-118.
  • [4] S. Kobayashi and N. Suita, On subordination of subharmonic functions, Kodai Math. J. 3 (1980), 315-320. MR 588461 (81m:31005)
  • [5] W. Rudin, Analytic functions of class $ {H_p}$, Trans. Amer. Math. Soc. 78 (1955), 46-66. MR 0067993 (16:810b)
  • [6] J. V. Ryff, Subordinate $ {H_p}$ functions, Duke Math. J. 33 (1966), 347-354. MR 0192062 (33:289)

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DOI: https://doi.org/10.1090/S0002-9939-1984-0740181-4
Article copyright: © Copyright 1984 American Mathematical Society

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