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Journal of Mathematical Sciences

, Volume 234, Issue 3, pp 381–383 | Cite as

Smoothness of a Holomorphic Function and Its Modulus on the Boundary of a Polydisk

  • N. A. ShirokovEmail author
Article
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We prove that if a function f is holomorphic in the polydisk 𝔻n, n ≥ 2, f is continuous in \( \overline{{\mathbb{D}}^n} \), f(z) ≠ 0, z ∈ 𝔻n, and |f| belongs to the α-Hölder class, 0 < α < 1, on the boundary of 𝔻n, then f belongs to the \( \left(\frac{\alpha }{2}-\varepsilon \right) \)-Hölder class on \( \overline{{\mathbb{D}}^n} \) for any ε > 0.

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References

  1. 1.
    N. A. Shirokov, “Smoothness of a function holomorphic in a ball and of its modulus on the sphere,” Zap. Nauchn. Semin. POMI, 447, 123–128 (2016).Google Scholar
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    N. A. Shirokov, “Hölder spaces in Lavrent’ev domains,” Zap. Nauchn. Semin. POMI, 282, 256–275 (2001).Google Scholar
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    W. Rudin, Function Theory in the Unit Ball ofn [Russian translation], Moscow (1984).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.St. Petersburg State University, St. Petersburg Branch of HSE UniversitySt. PetersburgRussia
  2. 2.St. Petersburg Department of the Steklov Mathematical InstituteSt. PetersburgRussia

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