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 Mat. Sb., 2017, Volume 208, Number 3, Pages 54–71 (Mi msb8802)

Holomorphic mappings of the unit disc into itself with two fixed points

V. V. Goryainov

Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region

Abstract: The paper is concerned with holomorphic mappings of the unit disc into itself with two fixed points. Two cases are considered: when one fixed point lies inside the disc and the other lies on the boundary and when both fixed points lie on the boundary. The effect that angular derivatives at boundary fixed points have on the properties of functions inside the unit disc is studied. Conditions on the angular derivatives to guarantee the existence of domains of univalence inside the unit disc are given. The effect of the angular derivatives on the values of the Taylor coefficients of functions is also examined.
Bibliography: 19 titles.

Keywords: holomorphic mapping, fixed point, domain of univalence, angular derivative, coefficient region.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00674-a This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 16-01-00674_a).

DOI: https://doi.org/10.4213/sm8802

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English version:
Sbornik: Mathematics, 2017, 208:3, 360–376

Bibliographic databases:

UDC: 517.54
MSC: Primary 30C55; Secondary 30C50, 30J99, 30C75

Citation: V. V. Goryainov, “Holomorphic mappings of the unit disc into itself with two fixed points”, Mat. Sb., 208:3 (2017), 54–71; Sb. Math., 208:3 (2017), 360–376

Citation in format AMSBIB
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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. V. V. Goryainov, “Holomorphic Mappings of a Strip into Itself with Bounded Distortion at Infinity”, Proc. Steklov Inst. Math., 298 (2017), 94–103
2. P. Gumenyuk, D. Prokhorov, “Value regions of univalent self-maps with two boundary fixed points”, Ann. Acad. Sci. Fenn. Math., 43:1 (2018), 451–462
3. D. V. Prokhorov, “Value regions in classes of conformal mappings”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 19:3 (2019), 258–279
4. O. S. Kudryavtseva, A. P. Solodov, “Two-sided estimate of univalence domains of holomorphic mappings of the disc into itself with an invariant diameter”, Russian Math. (Iz. VUZ), 63:7 (2019), 80–83
5. O. S. Kudryavtseva, A. P. Solodov, “Two-sided estimates for domains of univalence for classes of holomorphic self-maps of a disc with two fixed points”, Sb. Math., 210:7 (2019), 1019–1042
6. A. P. Solodov, “Strengthening of Landau's Theorem for Holomorphic Self-Mappings of a Disk with Fixed Points”, Math. Notes, 108:4 (2020), 626–628
7. O. S. Kudryavtseva, A. P. Solodov, “Asymptotically sharp two-sided estimate for domains of univalence of holomorphic self-maps of a disc with an invariant diameter”, Sb. Math., 211:11 (2020), 1592–1611
8. M. D. Contreras, S. Diaz-Madrigal, P. Gumenyuk, “Infinitesimal generators of semigroups with prescribed boundary fixed points”, Anal. Math. Phys., 10:3 (2020), 36
9. O. S. Kudryavtseva, “Schwarz's Lemma and Estimates of Coefficients in the Case of an Arbitrary Set of Boundary Fixed Points”, Math. Notes, 109:4 (2021), 653–657
10. V. N. Dubinin, “Some remarks on rotation theorems for complex polynomials”, Sib. elektron. matem. izv., 18:1 (2021), 369–376
11. O. S. Kudryavtseva, “Neravenstvo tipa Shvartsa dlya golomorfnykh otobrazhenii kruga v sebya s nepodvizhnymi tochkami”, Izv. vuzov. Matem., 2021, no. 7, 43–51
12. A. P. Solodov, “The exact domain of univalence on the class of holomorphic maps of a disc into itself with an interior and a boundary fixed points”, Izv. Math., 85:5 (2021), 1008–1035
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