Abstract
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.
Similar content being viewed by others
References
Z. Nehari, Conformal Mapping (Dover Publ., New York, 1975).
O. Lehto, Univalent Functions and Teichmülier Spaces, (Springer, New York, 1987).
B. Osgood, “Old and new on the Schwarzian derivative,” in Quasiconformal Mappings and Analysis (Springer, New York, 1998), pp. 275–308.
M. Chuaqui, P. Duren, W. Ma, D. Mejia, D. Minda, and B. Osgood, “Schwarzian norms and two-point distortion,” Pacific J. Math. 254 (1), 101–116 (2011).
V. N. Dubinin, “Geometric estimates for the Schwarzian derivative,” Uspekhi Mat. Nauk 72 (3(435)), 97–130 (2017) [Russian Math. Surveys 72 (3), 479–511 (2017)].
V. Bolotnikov, “Several inequalities for the Schwarzian derivative of a bounded analytic function,” Complex Var. Elliptic Equ. 64 (7), 1093–1102 (2019).
Z. Nehari, “The Schwarzian derivative and schlicht functions,” Bull. Amer. Math. Soc. 55, 545–551 (1949).
V. N. Dubinin, Condenser Capacities and Symmetrization in Geometric Function Theory (Springer, Basel, 2014).
M. Schiffer, “Some recent developments in the theory of conformal mapping,” in: R. Courant, Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces (New York, Interscience, 1950), pp. 249–323.
V. Singh, “Grunsky inequalities and coefficients of bounded schlicht functions,” in Ann. Acad. Sci. Fenn. Ser. A I (Suomalainen tiedeakatemia, Helsinki, 1962), Vol. 310.
Yu. E. Alenitsyn, “Univalent functions without common values in a multiply connected domain,” in Trudy Mat. Inst. Steklova, Vol. 94: Extremal Problems of the Geometric Theory of Functions (Nauka, Leningrad, 1968), pp. 4–18 [Proc. Steklov Inst. Math. 94, 1–18 (1968)].
V. N. Dubinin, “The majorization principle for p-valent functions,” Mat. Zametki 65 (4), 533–541 (1999) [Math. Notes 65 (4), 447–453 (1999)].
Funding
This work was carried out in the framework of the Russian State Assignment (grant no. 075-01095-20-00) and supported in part by the Russian Foundation for Basic Research under grant no. 20-01-00018.
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 6, pp. 865–872.
Rights and permissions
About this article
Cite this article
Dubinin, V.N. The Schwarzian Derivative of a p-Valent Function. Math Notes 107, 953–958 (2020). https://doi.org/10.1134/S0001434620050260
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434620050260