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

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

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
Related articles on Google Scholar: Russian articles, English articles

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
2. Jan Boman, “Equivalence of generalized moduli of continuity”, Ark Mat, 18:1-2 (1980), 73
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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
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
6. Raimo Näkki, Bruce Palka, “Extremal length and Hölder continuity of conformal mappings”, Comment Math Helv, 61:1 (1986), 389
7. P. M. Tamrazov, “Contour-solid results for holomorphic functions”, Math. USSR-Izv., 29:1 (1987), 193–205
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
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
10. E. P. Dolzhenko, “On boundary properties of the components of polyharmonic functions”, Math. Notes, 63:6 (1998), 724–735
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
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
13. Brigitte Forster, “Direct approximation theorems for Dirichlet series in the norm of uniform convergence”, Journal of Approximation Theory, 132:1 (2005), 1
14. Songxiao Li, Stevo Stević, “Volterra-Type Operators on Zygmund Spaces”, J Inequal Appl, 2007 (2007), 1
15. Songxiao Li, Stevo Stević, “Weighted composition operators from Zygmund spaces into Bloch spaces”, Applied Mathematics and Computation, 206:2 (2008), 825
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
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
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