Geodesic
Two-sided estimates for domains of univalence for classes of holomorphic self-maps of a disc with two fixed points
Sbornik. Mathematics, Tome 210 (2019) no. 7, pp. 1019-1042 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We investigate the class of holomorphic maps of a disc into itself that have an interior and a boundary fixed point, as well as the class of holomorphic maps of a half-plane into itself that have a fixed point in the interior of the domain and a fixed point at infinity. Two-sided estimates for domains of univalence are obtained for these function classes in terms of the values of the angular derivative at the boundary fixed point and the position of the interior fixed point. Bibliography: 21 titles.
Keywords: holomorphic map, fixed points, angular derivative
Mots-clés : domain of univalence. 
@article{SM_2019_210_7_a3,
     author = {O. S. Kudryavtseva and A. P. Solodov},
     title = {Two-sided estimates for domains of univalence for classes of holomorphic self-maps of a~disc with two fixed points},
     journal = {Sbornik. Mathematics},
     pages = {1019--1042},
     year = {2019},
     volume = {210},
     number = {7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2019_210_7_a3/}
}
TY  - JOUR
AU  - O. S. Kudryavtseva
AU  - A. P. Solodov
TI  - Two-sided estimates for domains of univalence for classes of holomorphic self-maps of a disc with two fixed points
JO  - Sbornik. Mathematics
PY  - 2019
SP  - 1019
EP  - 1042
VL  - 210
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/SM_2019_210_7_a3/
LA  - en
ID  - SM_2019_210_7_a3
ER  - 
%0 Journal Article
%A O. S. Kudryavtseva
%A A. P. Solodov
%T Two-sided estimates for domains of univalence for classes of holomorphic self-maps of a disc with two fixed points
%J Sbornik. Mathematics
%D 2019
%P 1019-1042
%V 210
%N 7
%U http://geodesic.mathdoc.fr/item/SM_2019_210_7_a3/
%G en
%F SM_2019_210_7_a3
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. Sbornik. Mathematics, Tome 210 (2019) no. 7, pp. 1019-1042. http://geodesic.mathdoc.fr/item/SM_2019_210_7_a3/

[1] G. M. Goluzin, Geometric theory of functions of a complex variable, Transl. Math. Monogr., 26, Amer. Math. Soc., Providence, RI, 1969, vi+676 pp. | DOI | MR | MR | Zbl | Zbl

[2] G. Valiron, Fonctions analytiques, Presses Universitaires de France, Paris, 1954, 236 pp. | MR | Zbl

[3] L. V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Math., McGraw-Hill Book Co., New York–Düsseldorf–Johannesburg, 1973, ix+157 pp. | MR | Zbl

[4] C. Carathéodory, “Über die Winkelderivierten von beschränkten analytischen Funktionen”, Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl., 1929 (1929), 39–54 | Zbl

[5] E. Landau, G. Valiron, “A deduction from Schwarz's lemma”, J. London Math. Soc., 4:3 (1929), 162–163 | DOI | MR | Zbl

[6] V. V. Goryainov, O. S. Kudryavtseva, “One-parameter semigroups of analytic functions, fixed points and the Koenigs function”, Sb. Math., 202:7 (2011), 971–1000 | DOI | DOI | MR | Zbl

[7] V. V. Goryainov, “Evolution families of conformal mappings with fixed points and the Löwner–Kufarev equation”, Sb. Math., 206:1 (2015), 33–60 | DOI | DOI | MR | Zbl

[8] V. V. Goryainov, “Holomorphic mappings of the unit disc into itself with two fixed points”, Sb. Math., 208:3 (2017), 360–376 | DOI | DOI | MR | Zbl

[9] F. G. Avkhadiev, L. A. Aksent'ev, “The main results on sufficient conditions for an analytic function to be schlicht”, Russian Math. Surveys, 30:4 (1975), 1–63 | DOI | MR | Zbl

[10] E. Study, Vorlesungen über ausgewälte Gegenstände der Geometrie. Zweiter Heft. Konforme Abbildung einfach zusammenhängender Bereiche, B. G. Teubner, Leipzig und Berlin, 1913, iv+142 pp. | Zbl

[11] J. W. Alexander, “Functions which map the interior of the unit circle upon simple regions”, Ann. of Math. (2), 17:1 (1915), 12–22 | DOI | MR | Zbl

[12] L. Špaček, “Příspěvek k teorii funkcí prostých [Contribution à la théorie des fonctions univalentes]”, Časopis Pěst. Mat. Fys., 62:2 (1932), 12–19 | Zbl

[13] Z. Nehari, “The Schwarzian derivative and schlicht functions”, Bull. Amer. Math. Soc., 55:6 (1949), 545–551 | DOI | MR | Zbl

[14] G. V. Kuzmina, “Chislennoe opredelenie radiusov odnolistnosti analiticheskikh funktsii”, Raboty po priblizhennomu analizu, Tr. MIAN SSSR, 53, Izd-vo AN SSSR, M.–L., 1959, 192–235 | MR | Zbl

[15] E. Landau, “Der Picard–Schottkysche Satz und die Blochsche Konstante”, Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl., 1926 (1926), 467–474 | Zbl

[16] Ch. Pommerenke, “On the iteration of analytic functions in a halfplane. I”, J. London Math. Soc. (2), 19:3 (1979), 439–447 | DOI | MR | Zbl

[17] J. Becker, Ch. Pommerenke, “Angular derivatives for holomorphic self-maps of the disk”, Comput. Methods Funct. Theory, 17:3 (2017), 487–497 | DOI | MR | Zbl

[18] C. Carathéodory, Conformal representation, Camb. Tracts Math. Math. Phys., 28, Reprint of 2nd. ed., Cambridge Univ. Press, Cambridge, 1969, x+115 pp. | MR | Zbl

[19] V. V. Prasolov, Polynomials, Algorithms Comput. Math., 11, Springer-Verlag, Berlin, 2004, xiv+301 pp. | DOI | MR | Zbl

[20] C. Pommerenke, Univalent functions, Studia Mathematica/Mathematische Lehrbücher, 25, Vandenhoeck Ruprecht, Göttingen, 1975, 376 pp. | MR | Zbl

[21] P. L. Duren, Univalent functions, Grundlehren Math. Wiss., 259, Springer-Verlag, New York, 1983, xiv+382 pp. | MR | Zbl