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Nonlinear Beltrami equation: lower estimates of Schwarz lemma’s type
Part of
Igor Petkov, Ruslan Salimov, Mariia Stefanchuk
Journal:

Canadian Mathematical Bulletin / Volume 67 / Issue 3 / September 2024

Published online by Cambridge University Press:

29 November 2023, pp. 533-543

Print publication:

September 2024
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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.