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Nonlinear Beltrami equation: lower estimates of Schwarz lemma’s type

Part of: Functions of several variables General properties Geometric function theory Classical measure theory

Published online by Cambridge University Press:  29 November 2023

Igor Petkov Open the ORCID record for Igor Petkov [Opens in a new window] ,
Ruslan Salimov Open the ORCID record for Ruslan Salimov [Opens in a new window]  and
Mariia Stefanchuk Open the ORCID record for Mariia Stefanchuk [Opens in a new window]
Igor Petkov
Affiliation:
Admiral Makarov National University of Shipbuilding, 9 Heroes of Ukraine Avenue, Mykolaiv 54007, Ukraine e-mail: igorpetkov83@gmail.com
Ruslan Salimov*
Affiliation:
Institute of Mathematics of NAS of Ukraine, 3 Tereschenkivska Street, Kyiv-4 01024, Ukraine e-mail: stefanmv43@gmail.com
Mariia Stefanchuk
Affiliation:
Institute of Mathematics of NAS of Ukraine, 3 Tereschenkivska Street, Kyiv-4 01024, Ukraine e-mail: stefanmv43@gmail.com
*
e-mail: ruslan.salimov1@gmail.com
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Abstract

We study a nonlinear Beltrami equation $f_\theta =\sigma \,|f_r|^m f_r$ in polar coordinates $(r,\theta ),$ which becomes the classical Cauchy–Riemann system under $m=0$ and $\sigma =ir.$ Using the isoperimetric technique, various lower estimates for $|f(z)|/|z|, f(0)=0,$ as $z\to 0,$ are derived under appropriate integral conditions on complex/directional dilatations. The sharpness of the above bounds is illustrated by several examples.

Keywords

Beltrami equationnonlinear Beltrami equationnonlinear Cauchy–Riemann systemasymptotic behaviorSchwarz lemma

MSC classification

Primary: 28A75: Length, area, volume, other geometric measure theory 30A10: Inequalities in the complex domain
Secondary: 26B15: Integration 30C80: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 3 , September 2024 , pp. 533 - 543
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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