The Schwarzian Derivative of a
Matematičeskie zametki, Tome 107 (2020) no. 6, pp. 865-872.

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

We show that a comparison of the capacities of suitable condensers gives an inequality for the Schwarzian derivatives of a holomorphic $p$-valent function defined on the unit disk and ranging in a given domain of the complex plane. If that domain is a disk as well and the function is univalent, then this inequality essentially coincides with the classical Nehari inequality. The cases of equality in the resulting relation are discussed.
Keywords: Schwarzian derivative, $p$-valent function, holomorphic function, condenser capacity, Green's function.
@article{MZM_2020_107_6_a5,
     author = {V. N. Dubinin},
     title = {The {Schwarzian} {Derivative} of a},
     journal = {Matemati\v{c}eskie zametki},
     pages = {865--872},
     publisher = {mathdoc},
     volume = {107},
     number = {6},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2020_107_6_a5/}
}
TY  - JOUR
AU  - V. N. Dubinin
TI  - The Schwarzian Derivative of a
JO  - Matematičeskie zametki
PY  - 2020
SP  - 865
EP  - 872
VL  - 107
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2020_107_6_a5/
LA  - ru
ID  - MZM_2020_107_6_a5
ER  - 
%0 Journal Article
%A V. N. Dubinin
%T The Schwarzian Derivative of a
%J Matematičeskie zametki
%D 2020
%P 865-872
%V 107
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2020_107_6_a5/
%G ru
%F MZM_2020_107_6_a5
V. N. Dubinin. The Schwarzian Derivative of a. Matematičeskie zametki, Tome 107 (2020) no. 6, pp. 865-872. http://geodesic.mathdoc.fr/item/MZM_2020_107_6_a5/

[1] Z. Nehari, Conformal Mapping, Dover Publ., New York, 1975 | MR

[2] O. Lehto, Univalent Fnctions and Teichmüller Spaces, Grad. Texts in Math., 109, Springer, New York, 1987 | MR

[3] B. Osgood, “Old and new on the Schwarzian derivative”, Quasiconformal Mappings and Analysis, Springer, New York, 1998, 275–308 | MR | Zbl

[4] M. Chuaqui, P. Duren, W. Ma, D. Me\'jia, D. Minda, B. Osgood, “Schwarzian norms and two-point distortion”, Pacific J. Math., 254:1 (2011), 101–116 | DOI | MR | Zbl

[5] V. N. Dubinin, “Geometricheskie otsenki proizvodnoi Shvartsa”, UMN, 72:3 (435) (2017), 97–130 | DOI | MR | Zbl

[6] V. Bolotnikov, “Several inequalities for the Schwarzian derivative of a bounded analytic function”, Complex Var. Elliptic Equ., 64:7 (2019), 1093–1102 | DOI | MR | Zbl

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

[8] V. N. Dubinin, Condenser Capacities and Symmetrization in Geometric Function Theory, Springer, Basel, 2014 | MR | Zbl

[9] M. Shiffer, “Nekotorye novye rezultaty v teorii konformnykh otobrazhenii”: R. Kurant, Prilozhenie, M., IL, 1953, 234–301

[10] V. Singh, Grunsky Inequalities and Coefficients of Bounded Schlicht Functions, Ann. Acad. Sci. Fenn. Ser. A I, 310, Suomalainen tiedeakatemia, Helsinki, 1962 | MR

[11] Yu. E. Alenitsyn, “Ob odnolistnykh funktsiyakh bez obschikh znachenii v mnogosvyaznoi oblasti”, Ekstremalnye zadachi geometricheskoi teorii funktsii, Tr. MIAN SSSR, 94, Nauka. Leningradskoe otdelenie, Leningrad, 1968, 4–18 | MR | Zbl

[12] V. N. Dubinin, “Printsip mazhoratsii dlya $p$-listnykh funktsii”, Matem. zametki, 65:4 (1999), 533–541 | DOI | MR | Zbl