Nonlinear Beltrami equation: lower estimates of Schwarz lemma's type.

Petkov, Igor; Salimov, Ruslan; Stefanchuk, Mariia

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.

ISOPERIMETRICAL problems; GEOMETRY; ISOPERIMETRIC inequalities; MATHEMATICAL inequalities; PLANE geometry

Canadian Mathematical Bulletin, 2024, Vol 67, Issue 3, p533

0008-4395

Academic Journal

10.4153/S0008439523000942