Geodesic
Avkhadiev--Lehto type constants in the study of the Gakhov class
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2021), pp. 47-55.

Voir la notice de l'article provenant de la source Math-Net.Ru

Avkhadiev’s classes (of holomorphic functions with two-sided bounds of the modulus of the derivative) are studied in domains other than a unit disk. We give the conditions that ensure the uniqueness of the critical point of the conformal radius for the images of the mentioned domains under the mappings by the functions of the Avkhadiev classes. We use an analogue of the setting proposed at the time by O. Lehto to study the univalence of functions satisfying the conditions of the Nehari type in domains conformally equivalent to a disk.
Keywords: conformal (inner mapping) radius, Avkhadiev's classes, regular Gakhov class
Mots-clés : Avkhadiev-Lehto type constants. 
@article{IVM_2021_3_a3,
     author = {A. V. Kazantsev and M. I. Kinder},
     title = {Avkhadiev--Lehto type constants in the study of the {Gakhov} class},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {47--55},
     publisher = {mathdoc},
     number = {3},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2021_3_a3/}
}
TY  - JOUR
AU  - A. V. Kazantsev
AU  - M. I. Kinder
TI  - Avkhadiev--Lehto type constants in the study of the Gakhov class
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2021
SP  - 47
EP  - 55
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2021_3_a3/
LA  - ru
ID  - IVM_2021_3_a3
ER  - 
%0 Journal Article
%A A. V. Kazantsev
%A M. I. Kinder
%T Avkhadiev--Lehto type constants in the study of the Gakhov class
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2021
%P 47-55
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2021_3_a3/
%G ru
%F IVM_2021_3_a3
A. V. Kazantsev; M. I. Kinder. Avkhadiev--Lehto type constants in the study of the Gakhov class. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2021), pp. 47-55. http://geodesic.mathdoc.fr/item/IVM_2021_3_a3/

[1] Avkhadiev F. G., Aksentev L. A., Elizarov A. M., Kazantsev A. V., “Issledovanie odnoznachnoi razreshimosti kraevykh zadach so svobodnymi parametrami metodami geometricheskoi teorii funktsii”, Materialy IV Mezhdunarodn. konf. «Lavrentevskie chteniya po matematike, mekhanike i fizike» (Kazan, 3–7 iyulya, 1995), Novosibirsk, 1995, 21

[2] Lavrentev M. A., Shabat B. V., Metody teorii funktsii kompleksnogo peremennogo, Nauka, M., 1973 | MR

[3] Gakhov F. D., Kraevye zadachi, Nauka, M., 1977

[4] Tumashev G. G., Nuzhin M. T., Obratnye kraevye zadachi i ikh prilozheniya, Kazan, 1965

[5] Fridman A., Variatsionnye printsipy i zadachi so svobodnymi granitsami, Nauka, M., 1990

[6] Avkhadiev F. G., “Obratnaya kraevaya zadacha dlya funktsii s osobennostyami”, Tr. semin. po kraev. zadacham, 21, Kazan, 1984, 5–19 | Zbl

[7] Avkhadiev F. G., Shabalina S. B., “Nuli koeffitsientov preobrazovanii i usloviya razreshimosti obratnykh kraevykh zadach”, Izv. vuzov. Matem., 1994, no. 8, 3–10 | MR

[8] Avkhadiev F. G., Aksentev L. A., Elizarov A. M., “Dostatochnye usloviya konechnolistnosti analiticheskikh funktsii i ikh prilozheniya”, Itogi nauki i tekhn. Ser. Matem. anal., 25, VINITI, M., 1987, 3–121 | MR

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

[10] Avkhadiev F. G., Konformnye otobrazheniya i kraevye zadachi, Izd-vo Kazansk. un-ta, Kazan, 2019

[11] Gehring F. W., Pommerenke Ch., “On the Nehari univalence criterion and quasicircles”, Comment. Math. Helv., 59 (1984), 226–242 | DOI | MR | Zbl

[12] Aksentev L. A., Kazantsev A. V., “Novoe svoistvo klassa Nekhari i ego primenenie”, Izv. vuzov. Matem., 1989, no. 8, 69–72 | Zbl

[13] Haegi H. R., “Extremalprobleme und Ungleichungen konformer Gebietsgrößen”, Compositio Math., 8:2 (1950), 81–111 | MR | Zbl

[14] Kazantsev A. V., “Mnozhestvo Gakhova v prostranstve Khornicha pri blokhovskikh ogranicheniyakh na predshvartsiany”, Uchen. zap. Kazansk. un-ta. Ser. Fiz.-matem. nauki, 155, no. 2, 2013, 65–82 | MR | Zbl

[15] Kazantsev A. V., “O vykhode iz mnozhestva Gakhova, kontroliruemom usloviyami podchinennosti”, Uchen. zap. Kazansk. un-ta. Ser. Fiz.-matem. nauki, 156, no. 1, 2014, 31–43

[16] Ruscheweyh St., Wirths K.-J., “On extreme Bloch functions with prescribed critical points”, Math. Z., 180 (1982), 91–106 | DOI | MR

[17] Aksentev L. A., “Svyaz vneshnei obratnoi kraevoi zadachi s vnutrennim radiusom oblasti”, Izv. vuzov. Matem., 1984, no. 2, 3–11 | MR | Zbl

[18] Lehto O., “Domain constants associated with Schwarzian derivative”, Comment. Math. Helv., 52:4 (1977), 603–610 | DOI | MR | Zbl

[19] Lehto O., “Univalent functions, Schwarzian derivatives and quasiconformal mappings”, L'Enseignement Math., 24:3–4 (1978), 203–214 | MR | Zbl

[20] Kazantsev A. V., “O vykhode iz mnozhestva Gakhova po semeistvu klassov Avkhadieva”, Uchen. zap. Kazan. un-ta., Ser. Fiz.-matem. nauki, 159, no. 3, 2017, 318–326 | MR

[21] Goluzin G. M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966 | MR

[22] Bakelman I. Ya., Verner A. L., Kantor B. E., Vvedenie v diferentsialnuyu geometriyu «v tselom», Nauka, M., 1973 | MR

[23] Kazantsev A. V., “Bifurkatsii i novye usloviya edinstvennosti kriticheskikh tochek giperbolicheskikh proizvodnykh”, Uchen. zap. Kazansk. un-ta. Ser. Fiz.-matem. nauki, 153, no. 1, 2011, 180–194 | MR | Zbl

[24] Kazantsev A. V., Chetyre etyuda na temu F. D. Gakhova, Mariisk. un-t, Ioshkar-Ola, 2012