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.

中文翻译：

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非线性贝尔特拉米方程：施瓦茨引理类型的较低估计

我们研究极坐标$(r,\theta ),$中的非线性 Beltrami 方程$f_\theta =\sigma \,|f_r|^m f_r$ ，它在$m=0$和$ \sigma =ir.$使用等周技术，$|f(z)|/|z|, f(0)=0,$ 的各种较低估计值作为$z\to 0,$在适当的积分条件下导出复杂/定向扩张。上述界限的尖锐性可以通过几个例子来说明。