This article is cited in 9 papers

On the Schwarzian norm of harmonic mappings

S. Yu. Graf<sup>ab

a</sup> Petrozavodsk State University, 33, Lenina st., 185910 Petrozavodsk, Russia
^b Tver State University, 33, Zheliabova st., 170100 Tver, Russia

Abstract: We obtain estimations of the pre-Schwarzian and Schwarzian derivatives in terms of the order of family in linear and affine invariant families $\mathcal{L}$ of sense preserving harmonic mappings of the unit disk $\mathbb{D}$. As the converse result the order of family $\mathcal{L}$ is estimated in terms of suprema of Schwarzian and pre-Schwarzian norms over the family $\mathcal{L}$. Main results are obtained by means of theory of linear invariant families.

Keywords: harmonic mappings; linear invariant families; Schwarzian derivative.

UDC: 517.54

MSC: 30C45, 30C55, 30C99

Received: 19.10.2016
Revised: 11.12.2016
Accepted: 13.12.2016

Language: English

DOI: 10.15393/j3.art.2016.3511

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