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Uspekhi Mat. Nauk, 1973, Volume 28, Issue 1(169), Pages 131–161 (Mi umn4836)  

This article is cited in 17 scientific papers (total in 17 papers)

Contour and solid structure properties of holomorphic functions of a complex variable

P. M. Tamrazov


Abstract: For a $f$ function holomorphic in an open set $G$ the paper solves problems on the relationships between its properties along $\partial G$, the boundary of $G$, on the one hand and along $\overline G$, the closure of $G$, on the other. The properties discussed are those that can be expressed in terms of the derivatives, moduli of continuity, and rates of decrease or increase of the function along $\overline G$ and along $\partial G$. The results are established for very wide classes of sets $G$ and majorants of the moduli of continuity. In particular, all the main results are true for every bounded simply-connected domain and any majorant of the type of a modulus of continuity. A number of problems posed in 1942 by Sewell are solved.

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English version:
Russian Mathematical Surveys, 1973, 28:1, 141–173

Bibliographic databases:

UDC: 517.54
MSC: 32A10, 30G12
Received: 10.01.1972

Citation: P. M. Tamrazov, “Contour and solid structure properties of holomorphic functions of a complex variable”, Uspekhi Mat. Nauk, 28:1(169) (1973), 131–161; Russian Math. Surveys, 28:1 (1973), 141–173

Citation in format AMSBIB
\Bibitem{Tam73}
\by P.~M.~Tamrazov
\paper Contour and solid structure properties of holomorphic functions of a~complex variable
\jour Uspekhi Mat. Nauk
\yr 1973
\vol 28
\issue 1(169)
\pages 131--161
\mathnet{http://mi.mathnet.ru/umn4836}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=409821}
\zmath{https://zbmath.org/?q=an:0256.30036|0273.30036}
\transl
\jour Russian Math. Surveys
\yr 1973
\vol 28
\issue 1
\pages 141--173
\crossref{https://doi.org/10.1070/RM1973v028n01ABEH001398}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. L.A Rubel, A.L Shields, B.A Taylor, “Mergelyan sets and the modulus of continuity of analytic functions”, Journal of Approximation Theory, 15:1 (1975), 23  crossref
    2. Jan Boman, “Equivalence of generalized moduli of continuity”, Ark Mat, 18:1-2 (1980), 73  crossref  mathscinet  zmath  isi
    3. F.W Gehring, W.K Hayman, A Hinkkanen, “Analytic functions satisfying Hölder conditions on the boundary”, Journal of Approximation Theory, 35:3 (1982), 243  crossref
    4. B. Jöricke, “The relation between the solid modulus of continuity and the modulus of continuity along the Shilov boundary for analytic functions of several variables”, Math. USSR-Sb., 50:2 (1985), 495–511  mathnet  crossref  mathscinet  zmath
    5. V. V. Andrievskii, “The geometric structure of regions, and direct theorems of the constructive theory of functions”, Math. USSR-Sb., 54:1 (1986), 39–56  mathnet  crossref  mathscinet  zmath
    6. Raimo Näkki, Bruce Palka, “Extremal length and Hölder continuity of conformal mappings”, Comment Math Helv, 61:1 (1986), 389  crossref  mathscinet  zmath  isi
    7. P. M. Tamrazov, “Contour-solid results for holomorphic functions”, Math. USSR-Izv., 29:1 (1987), 193–205  mathnet  crossref  mathscinet  zmath
    8. F. A. Shamoyan, “Closed ideals in algebras of functions analytic in the disk and smooth up to its boundary”, Russian Acad. Sci. Sb. Math., 79:2 (1994), 425–445  mathnet  crossref  mathscinet  zmath  isi
    9. E. P. Dolzhenko, “Some remarks on the modulus of continuity of a conformal mapping of the disk onto a Jordan domain”, Math. Notes, 60:2 (1996), 130–136  mathnet  crossref  crossref  mathscinet  zmath  isi
    10. E. P. Dolzhenko, “On boundary properties of the components of polyharmonic functions”, Math. Notes, 63:6 (1998), 724–735  mathnet  crossref  crossref  mathscinet  zmath  isi
    11. Brigitte Forster, “On the Relation Between Fourier and Leont’ev Coefficients with Respect to the Space AC(D)”, Comput. Methods Funct. Theory, 1:1 (2002), 193  crossref
    12. A. V. Khaustov, N. A. Shirokov, “A converse approximation theorem on subsets of elliptic curves”, J. Math. Sci. (N. Y.), 133:6 (2006), 1756–1764  mathnet  crossref  mathscinet  zmath  elib  elib
    13. Brigitte Forster, “Direct approximation theorems for Dirichlet series in the norm of uniform convergence”, Journal of Approximation Theory, 132:1 (2005), 1  crossref
    14. Songxiao Li, Stevo Stević, “Volterra-Type Operators on Zygmund Spaces”, J Inequal Appl, 2007 (2007), 1  crossref  mathscinet  zmath  isi
    15. Songxiao Li, Stevo Stević, “Weighted composition operators from Zygmund spaces into Bloch spaces”, Applied Mathematics and Computation, 206:2 (2008), 825  crossref
    16. A. Yu. Timofeev, “Zadacha Dirikhle dlya golomorfnykh funktsii v prostranstvakh s zadannym modulem nepreryvnosti”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2011, no. 3, 107–116  mathnet
    17. E. P. Dolzhenko, “Bounds for the moduli of continuity for conformal mappings of domains near their accessible boundary arcs”, Sb. Math., 202:12 (2011), 1775–1823  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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