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Nonnormal Dirichlet quotients and nonnormal Blaschke quotients


Author: Shinji Yamashita
Journal: Proc. Amer. Math. Soc. 80 (1980), 604-606
MSC: Primary 30D45; Secondary 30D50
DOI: https://doi.org/10.1090/S0002-9939-1980-0587936-3
MathSciNet review: 587936
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Abstract: There exists a nonnormal meromorphic function $ {f_1}/{f_2}$ in $ U = \{ \vert z\vert < 1\} $, where $ {f_1}$ and $ {f_2}$ both are holomorphic functions with finite Dirichlet integrals in U. For each $ 0 < \alpha < 1$, there exists a nonnormal meromorphic function $ {B_1}/{B_2}$ in U, where $ {B_1}$ and $ {B_2}$ both are Blaschke products with finite $ \alpha $-weighted Dirichlet integrals in U.


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

  • [1] F. Bagemihl and W. Seidel, Sequential and continuous limits of meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I 280 (1960), 1-17. MR 0121488 (22:12226)
  • [2] J. A. Cima, A nonnormal Blaschke-quotient, Pacific J. Math. 15 (1965), 767-773. MR 0190344 (32:7757)
  • [3] P. L. Duren, Theory of $ {H^p}$ spaces, Academic Press, New York and London, 1970. MR 0268655 (42:3552)
  • [4] V. I. Gavrilov, On the theorems of Beurling, Carleson and Tsuji on exceptional sets, Mat. Sb. 94 (136) (1974), 3-15; English transl. in Math. USSR Sb. 23 (1974), 1-12. MR 0352467 (50:4954)
  • [5] O. Lehto and K. I. Virtanen, Boundary behaviour and normal meromorphic functions, Acta Math. 97 (1957), 47-65. MR 0087746 (19:403f)
  • [6] D. Protas, Blaschke products with derivative in $ {H^p}$ and $ {B^p}$, Michigan Math. J. 20 (1973), 393 - 396. MR 0344478 (49:9217)
  • [7] H. S. Shapiro and A. L. Shields, On the zeros of functions with finite Dirichlet integral and some related function spaces, Math. Z. 80 (1962), 217-229. MR 0145082 (26:2617)
  • [8] N. Yanagihara, On a quotient of functions with finite Dirichlet integrals, J. College Arts Sci. Chiba Univ. 4 (1966), no. 4, 395. MR 0218536 (36:1622)
  • [9] A. Zygmund, Trigonometric series. II, Cambridge Univ. Press, London and New York, 1959. MR 0107776 (21:6498)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0587936-3
Article copyright: © Copyright 1980 American Mathematical Society

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