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

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.

1 Introduction

Recently, numerous nonlinear counterparts of the classical Beltrami equation are successfully studied by many mathematicians. The directional dilatation coefficients provide fruitful and flexible tools for investigating the main features of regular solutions to these equations (see [2, 9, 26]).

In this paper, we estimate the ratio $|f(z)|/|z|$ under $f(0)=0,$ which can be treated as asymptotic dilation or real differentiability condition (cf. [6, 10]) at the origin and goes back to the classical Schwarz lemma bounds.

Let $\mathbb {C}$ be the complex plane. In the complex notation $f=u+iv$ and $z=x+iy$ , the Beltrami equation in a domain $G\subset \mathbb {C}$ has the form

(1.1) $$ \begin{align} f_{\overline{z}}\,=\,\mu(z)f_{z}, \end{align} $$

where $\mu \colon G\to \mathbb {C}$ is a measurable function and

$$ \begin{align*}f_{\overline{z}}=\frac{1}{2}(f_{x}+if_{y})\qquad \mathrm{and} \qquad f_{z}=\frac{1}{2}(f_{x}-if_{y})\end{align*} $$

are formal derivatives of f in $\overline {z}$ and z, while $f_{x}$ and $f_{y}$ are partial derivatives of f in the variables x and y, respectively.

Various existence theorems for solutions of the Sobolev class have been recently established applying the modulus approach for a quite wide class of linear and quasilinear degenerate Beltrami equations (see, e.g., [14, 15, 23, 26, 34, 35]).

Let $\sigma \colon G \to \mathbb {C}$ be a measurable function and $m\geqslant 0.$ We consider the following equation written in the polar coordinates $(r,\theta )$ :

(1.2) $$ \begin{align} f_{r}=\sigma(re^{i\theta})\,|f_{\theta}|^{m}\, f_{\theta}, \end{align} $$

where $f_{\theta }$ and $f_{r}$ are the partial derivatives of f by $\theta $ and $r,$ respectively. The equations of this type were studied in the works [11, 12, 30–33].

Applying the relations between these derivatives and the formal derivatives

(1.3) $$ \begin{align} rf_{r}\,=\,zf_{z}+\overline{z}f_{\overline{z}}\,, \qquad f_{\theta}\,=\,i(zf_{z}-\overline{z}f_{\overline{z}}) \end{align} $$

(see, e.g., [5, (21.51)]), one can rewrite equation (1.2) in the Cartesian form:

(1.4) $$ \begin{align} f_{\overline{z}}\,=\,\frac{z}{\overline{z}}\,\frac{\widetilde{\sigma}(z)\,|zf_{z}-\overline{z}f_{\overline{z}}|^{m}-1}{\widetilde{\sigma}(z)\,|zf_{z}-\overline{z}f_{\overline{z}}|^{m}+1}\, f_{z}\,, \end{align} $$

where $\widetilde {\sigma }(z)=i\sigma (z)|z|$ .

Under $m=0$ , equation (1.4) reduces to the standard linear Beltrami equation (1.1) with the complex coefficient

$$ \begin{align*} \mu(z)\,=\,\frac{z}{\overline{z}}\,\frac{i\sigma(z)\,|z|-1}{i\sigma(z)\,|z|+1}\,. \end{align*} $$

Picking $m=0$ and $\sigma =-i/|z|$ in (1.4), we arrive at the classical Cauchy–Riemann system. For $m>0$ , equation (1.4) provides a partial case of the general nonlinear system of equations (7.33) given in [5, Section 7.7].

Next, we consider an equation of another type, namely

(1.5) $$ \begin{align} f_{\theta}=\sigma(re^{i\theta})\,|f_{r}|^{m}\, f_{r}. \end{align} $$

Applying the relations (1.3), one can rewrite equation (1.5) by

(1.6) $$ \begin{align} f_{\overline{z}}\,=\,\frac{z}{\overline{z}}\,\frac{1+i\sigma(z)\,|z|^{-m-1}|zf_{z}+\overline{z}f_{\overline{z}}|^{m}}{1-i\sigma(z)\,|z|^{-m-1}|zf_{z}+\overline{z}f_{\overline{z}}|^{m}}\, f_{z}. \end{align} $$

Assuming $m=0$ , equation (1.6) also becomes the standard linear Beltrami equation (1.1) with

$$ \begin{align*} \mu(z)\,= \,\frac{z}{\overline{z}}\,\frac{1+i\sigma(z)/|z|}{1-i\sigma(z)/|z|}\, \,. \end{align*} $$

Choosing $m=0$ and $\sigma =i|z|$ in (1.6), we arrive again at the classical Cauchy–Riemann system. Later on, we assume that $m>0.$

A mapping $f \colon G \to \mathbb C$ is called regular at a point $z_0\in G,$ if f has the total differential at this point and its Jacobian $J_f=|f_z|^{2}-|f_{\bar {z}}|^{2}$ does not vanish (cf. [22, I.1.6]). A homeomorphism f of Sobolev class $W_{\mathrm {{loc}}}^{1, 1}$ is called regular, if $J_{f}>0$ a.e. By a regular solution of equation (1.6), we call a regular homeomorphism $f \colon G \to \mathbb C,$ which satisfies (1.6) a.e. in $G.$

The nonlinear equations (1.4) and (1.6) provide partial cases of the nonlinear system of two real partial differential equations (see [21, (1)] and [19, 20]). Note that various nonlinear systems of similar partial differential equations studied from different aspects and features can be found in [1, 3–5, 7, 8, 13, 17–21, 25, 27, 28].

2 Auxiliary results

Later on, we use the following notations:

$$ \begin{align*} B_r=\{z\in\mathbb C: |z|< r\}\,, \quad \mathbb B=\{z\in\mathbb C: |z| < 1\} \end{align*} $$

and

$$ \begin{align*} \gamma_r=\{z\in\mathbb C: |z|=r\}\,,\, \ \ \mathbb{A}(0,r_1,r_2)=\{z\in \mathbb{C}: \, r_1<|z|<r_2 \}. \end{align*} $$

The area of set $f(B_r)$ we denote by $S_f(r)=|f(B_r)|.$

Let $f:\mathbb {B}\to \mathbb {C}$ be a regular homeomorphism of the Sobolev class $W_{\mathrm {loc }}^{1,1}$ , and let $p>1$ . By the p-angular dilatation of the mapping f with respect to the point $z_0=0$ , we call a quantity

$$ \begin{align*}D_{p,f}(z) = D_{p,f}(re^{i\theta}) = \frac{|f_{\theta}(re^{i\theta})|^p}{r^p J_f(re^{i\theta})}\,,\end{align*} $$

where $z = re^{i\theta }$ and $J_f$ is the Jacobian of f.

For $D_{p,f}(z)$ and $p>1$ , denote

(2.1) $$ \begin{align} d_{p,f}(r)=\left( \frac{1}{2\pi r} \int\limits_{\gamma_r}\, D_{p,f}^{\frac{1}{p-1}}(z) \, |dz| \right)^{p-1}\,. \end{align} $$

The following statement provides a differential inequality for the area functional $S_f(r)=|f(B_r)|$ (see Lemma 2.1 in [12]).

Proposition 2.1 Let $f:\mathbb {B}\to \mathbb {C}$ be a regular homeomorphism of the Sobolev class $W_{\mathrm {loc }}^{1,1}$ that possesses the N-property, and let $p> 1$ . Then

$$ \begin{align*}S_f'(r) \geqslant 2\pi^{\frac{2-p}{2}} r^{1-p} d_{p,f}^{-1}(r) S_f^{\frac{p}{2}}(r)\end{align*} $$

for almost all (a.a.) $r \in [0, 1)$ .

Proposition 2.2 Let $f:\mathbb {B}\to \mathbb {C}$ be a regular homeomorphism of the Sobolev class $W_{\mathrm {loc }}^{1,1}$ that possesses the N-property, and let $p> 1$ and $K>0$ . If

(2.2) $$ \begin{align} d_{p,f}(r)\leqslant K \quad \text{for a.a.} \quad r\in (0,1)\,, \end{align} $$

then

(2.3) $$ \begin{align} S'_f(r)\geqslant 2\pi^{\frac{2-p}{2}}\, K^{-1}\, r^{1-p}\, S_f^{\frac{p}{2}}(r) \end{align} $$

for a.a. $r\in [0,1).$

Proof Indeed, due to the condition (2.2) and applying Proposition 2.1, we obtain (2.3).

Lemma 2.3 Let $f:\mathbb {B}\to \mathbb {C}$ be a regular homeomorphism of the Sobolev class $W_{\mathrm {loc }}^{1,1}$ that possesses the N-property, $1<p <2$ , and $K>0$ . If $d_{p,f}(r)\leqslant K$ for a.a. $r\in (0,1)$ , then for $r\in [0,1)$ ,

(2.4) $$ \begin{align} |f(B_r)|\geqslant\, C(p,K)\, r^2\,, \end{align} $$

where $ C(p,K)=\pi K^{\frac {2}{p-2}} $ .

Proof Let $0<\varepsilon <r<1$ . By (2.3), for a.a. $t\in [0,1)$ ,

$$ \begin{align*}\frac{S'_f(t)\, dt }{S^{\frac{p}{2}}_f(t)}\geqslant 2\pi^{\frac{2-p}{2}}\, K^{-1}\, t^{1-p}\,dt, \end{align*} $$

and integrating over the segment $[\varepsilon , r]$ , we obtain

$$ \begin{align*}\int\limits_\varepsilon^r \frac{S'_f(t) }{S^{\frac{p}{2}}_f(t)} \, dt \geqslant 2\pi^{\frac{2-p}{2}}\, K^{-1}\, \int\limits_\varepsilon^r t^{1-p}\,dt\,. \end{align*} $$

Hence,

$$ \begin{align*}\int\limits_\varepsilon^r \phi'_p(t)\, dt \geqslant \frac{2\pi^{\frac{2-p}{2}}\,}{(2-p)K} \left(r^{2-p}-\varepsilon^{2-p}\right), \end{align*} $$

where $\phi _p(t)=\frac {2}{2-p}\, S^{\frac {2-p}{2}}_f(t)\,.$

This function is nondecreasing on $[0, 1)$ , and

$$ \begin{align*}\int\limits_\varepsilon^r \phi'_p(t)\, dt \leqslant \phi_p(r)- \phi_p(\varepsilon)\,=\frac{2}{2-p}\, \left(S^{\frac{2-p}{2}}_f(r)-S^{\frac{2-p}{2}}_f(\varepsilon)\right) \end{align*} $$

(see Theorem IV.7.4 in [29]). Combining the last two inequalities, we have

$$ \begin{align*}S^{\frac{2-p}{2}}_f(r)\geqslant S^{\frac{2-p}{2}}_f(r)-S^{\frac{2-p}{2}}_f(\varepsilon) \geqslant \frac{\pi^{\frac{2-p}{2}}\,}{K} \left(r^{2-p}-\varepsilon^{2-p}\right)\,. \end{align*} $$

Finally, letting $\varepsilon \to 0$ , we get the estimate (2.4).

The following result is an analog of the well-known Ikoma–Schwartz lemma on estimating the limsup (see Corollary 3 in [16]).

Lemma 2.4 Let $f: \mathbb {B}\to \mathbb {C}$ be a regular homeomorphism of the Sobolev class $W_{\mathrm {loc }}^{1,1}$ that possesses the N-property and normalized by $f(0)= 0$ , and $1<p <2$ and $K>0$ . If $d_{p,f}(r)\leqslant K$ for a.a. $r\in (0,1)$ , then

$$ \begin{align*}\limsup\limits_{z\to 0} \, \frac{|f(z)|}{|z|}\geqslant K^{-\frac{1}{2-p}}\,.\end{align*} $$

Proof Denote $\mathcal {M}_f(r)=\max \limits _{|z|=r}|f(z)|$ . Since $f(0)=0$ , we have

(2.5) $$ \begin{align} S_f(r)\leqslant \pi \, \mathcal{M}^2_f(r)\,. \end{align} $$

Thus, applying Lemma 2.3, we obtain

$$ \begin{align*}\limsup\limits_{z\to 0} \, \frac{|f(z)|}{|z|}=\limsup\limits_{r\to 0} \frac{\mathcal{M}_f(r)}{r}\geqslant \limsup\limits_{r\to 0} \sqrt{\frac{S_f(r)}{\pi r^2}}\geqslant K^{-\frac{1}{2-p}}\,.\\[-43pt] \end{align*} $$

In particular, by Lemma 2.4, we come to the following statement.

Corollary 2.5 If $K>0$ and $D_{p,f}(z)\leqslant K$ for a.a. $z\in \mathbb {B}$ , then $ \limsup \limits _{z\to 0} \, \frac {|f(z)|}{|z|}\geqslant K^{-\frac {1}{2-p}}\,.$

Next, we prove the following statement.

Theorem 2.6 Let $f: \mathbb {B}\to \mathbb {C}$ be a regular homeomorphism of the Sobolev class $W_{\mathrm {loc }}^{1,1}$ that possesses the N-property and normalized by $f(0)= 0$ , and let $1<p <2$ . Suppose that

$$ \begin{align*}\kappa_0=\liminf\limits_{\varepsilon\to 0} \left( \frac{1}{\pi\varepsilon^2}\int\limits_{B_{\varepsilon}}D_{p,f}^ {\frac{1}{p-1}}(z)\,dx\,dy\right)^{p-1}\,. \end{align*} $$

1) If $\kappa _0\in (0, \infty )$ , then

$$ \begin{align*}\limsup\limits_{z\to 0} \, \frac{|f(z)|}{|z|}\geqslant c_p \, \kappa_0^{-\frac{1}{2-p}}\,,\end{align*} $$

where $c_p$ is a positive constant depending on the parameter p.

2) If $\kappa _0=0$ , then

$$ \begin{align*}\limsup\limits_{z\to 0} \, \frac{|f(z)|}{|z|}=\infty\,.\end{align*} $$

Proof Let $\varepsilon \in \left (0, 1 \right )$ . By Proposition 2.1, for a.a. $r\in [0,1)$ ,

$$ \begin{align*}\frac{S'_f(r)\, dr }{S^{\frac{p}{2}}_f(r)}\geqslant \frac{2\pi^{\frac{2-p}{2}}}{r^{p-1}d_{p,f}(r) }\,dr; \end{align*} $$

therefore, integrating over the segment $[\frac {\varepsilon }{2}, \varepsilon ]$ , and arguing similarly to the proof of Lemma 2.3, we reach

$$ \begin{align*}S^{\frac{2-p}{2}}_f(\varepsilon)\geqslant S^{\frac{2-p}{2}}_f(\varepsilon)-S^{\frac{2-p}{2}}_f\left(\frac{\varepsilon}{2}\right) \geqslant \pi^{\frac{2-p}{2}}(2-p) \int\limits_{\frac{\varepsilon}{2}}^{\varepsilon} \frac{dr}{r^{p-1}d_{p,f}(r) }. \end{align*} $$

Hence,

(2.6) $$ \begin{align} S_f(\varepsilon)\geqslant \pi(2-p)^{\frac{2}{2-p}} \left(\int\limits_{\frac{\varepsilon}{2}}^{\varepsilon} \frac{dr}{r^{p-1}d_{p,f}(r) } \right)^{\frac{2}{2-p}} \,. \end{align} $$

Noting

$$ \begin{align*}\frac{\varepsilon}{2}=\int\limits_{\frac{\varepsilon}{2}}^{\varepsilon} \frac{1}{\left(\int\limits_{\gamma_{r}}D_{p,f}^{\frac{1}{p-1}}(z)\,|dz|\right)^{\frac{p-1}{p}}} \left(\int\limits_{\gamma_{r}}D_{p,f}^{\frac{1}{p-1}}(z)\,|dz|\right)^{\frac{p-1}{p}}dr, \end{align*} $$

one obtains by Holder’s inequality with exponents p and $p^{'}=p/(p-1),$

$$ \begin{align*}\left(\frac{\varepsilon}{2}\right)^{p}\leqslant\int\limits_{\frac{\varepsilon}{2}}^{\varepsilon} \frac{dr}{\left(\int\limits_{\gamma_{r}}D_{p,f}^{\frac{1}{p-1}}(z)\,|dz|\right)^{p-1}} \left(\int\limits_{\frac{\varepsilon}{2}}^{\varepsilon} \int\limits_{\gamma_{r}}D_{p,f}^{\frac{1}{p-1}}(z)\,|dz|\,dr\right)^{p-1}. \end{align*} $$

Due to (2.1), we rewrite the last inequality in the following form:

$$ \begin{align*}\left(\frac{\varepsilon}{2}\right)^{p}\leqslant \frac{1}{(2\pi)^{p-1}} \int\limits_{\frac{\varepsilon}{2}}^{\varepsilon}\frac{dr}{r^{p-1}d_{p,f}(r)} \left(\int\limits_{\frac{\varepsilon}{2}}^{\varepsilon}\int\limits_{\gamma_{r}}D_{p,f}^{\frac{1}{p-1}}(z)\,|dz|\,dr\right)^{p-1}. \end{align*} $$

Now, by the Fubini theorem, we have

$$ \begin{align*}\int\limits_{\frac{\varepsilon}{2}}^{\varepsilon}\frac{dr}{r^{p-1}d_{p,f}(r)}\geqslant (2\pi)^{p-1}\left(\frac{\varepsilon}{2}\right)^p\left(\, \int\limits_{A(0, \frac{\varepsilon}{2}, \varepsilon)}D_{p,f}^ {\frac{1}{p-1}}(z)\,dx\,dy\right)^{1-p}\,. \end{align*} $$

Since $ A(0, \frac {\varepsilon }{2}, \varepsilon )\subset B_{\varepsilon }$ ,

(2.7) $$ \begin{align} \int\limits_{\frac{\varepsilon}{2}}^{\varepsilon}\frac{dr}{r^{p-1}d_{p,f}(r)}\geqslant (2\pi)^{p-1}\left(\frac{\varepsilon}{2}\right)^p\left( \int\limits_{B_{\varepsilon}}D_{p,f}^ {\frac{1}{p-1}}(z)\,dx\,dy\right)^{1-p}. \end{align} $$

Combining (2.6) and (2.7), we obtain

$$ \begin{align*}S_f(\varepsilon)\geqslant \pi\left( \frac{2-p}{2} \right)^{\frac{2}{2-p}} \varepsilon^2 \left(\frac{1}{\pi\varepsilon^2}\int\limits_{B_{\varepsilon}}D_{p,f}^ {\frac{1}{p-1}}(z)\,dx\,dy\right)^{-\frac{2(p-1)}{2-p}} \,, \end{align*} $$

and by (2.5),

$$ \begin{align*}\frac{\mathcal{M}_f(\varepsilon)}{\varepsilon}\geqslant\sqrt{\frac{S_f(\varepsilon)}{\pi\varepsilon^2}}\geqslant c_p \left( \frac{1}{\pi\varepsilon^2}\int\limits_{B_{\varepsilon}}D_{p,f}^ {\frac{1}{p-1}}(z)\,dxdy\right)^{-\frac{p-1}{2-p}} \,, \end{align*} $$

where $c_p=\left (\frac {2-p}{2}\right )^{\frac {1}{2-p}}$ .

Thus,

$$ \begin{align*}\limsup\limits_{z\to 0} \, \frac{|f(z)|}{|z|}=\limsup\limits_{\varepsilon\to 0} \frac{\mathcal{M}_f(\varepsilon)}{\varepsilon}\geqslant c_p \, \kappa_0^{-\frac{1}{2-p}}. \end{align*} $$

This completes the proof of Theorem 2.6.

3 Application to nonlinear Beltrami equations

In this section, we present theorems on the asymptotic behavior of regular homeomorphic solutions to a nonlinear Beltrami equation of the form (1.6).

Theorem 3.1 Let $f: \mathbb {B}\to \mathbb {C}$ be a regular homeomorphic solution of equation (1.6) which belongs to Sobolev class $W_{\mathrm {loc }}^{1,2}$ , and normalized by $f(0) = 0$ . Assume that $C>0$ and the coefficient $\sigma : \mathbb {B}\to \mathbb {C}$ satisfies the following condition:

(3.1) $$ \begin{align} \int\limits_{\gamma_r} \frac{|\sigma(z) |^{m+2} }{\, \left(\mathrm{Im}\,\sigma(z)\right)^{m+1} }\, |dz| \leqslant C\, r^{2} \end{align} $$

for a.a. $r\in (0,1)$ . Then

(3.2) $$ \begin{align} \limsup\limits_{z\to 0} \, \frac{|f(z)|}{|z|}\geqslant \left(\frac{2\pi}{C}\right)^{\frac{1}{m}}\,. \end{align} $$

Proof Let $p=\frac {m+2}{m+1}$ , $1<p<2$ . It is well known that any homeomorphism $W_{\mathrm {loc }}^{1,2}$ possesses the N-property (see [24, Theorem 4.4]). Since f is a regular homeomorphic solution of equation (1.6), we get

$$ \begin{align*}J_f(re^{i\theta})= \frac{1}{r} \, \mathrm{Im}\, \left(\overline{f_r}\, f_\theta \right)=\frac{1}{r} \, |f_r|^{m+2} \, \mathrm{Im}\, \sigma(re^{i\theta})> 0 \ \text {a.e.}\end{align*} $$

and

$$ \begin{align*}D_{p,f}(re^{i\theta}) = \frac{|f_{\theta}(re^{i\theta})|^p}{r^p J_f(re^{i\theta})}= \frac{|\sigma (re^{i\theta}) |^{\frac{m+2}{m+1}} }{r^{\frac{1}{m+1}} \, \mathrm{Im}\,\sigma (re^{i\theta})}\,. \end{align*} $$

Therefore, in view of the condition (3.1), we have

$$ \begin{align*}d_{p,f}(r)= \left( \frac{1}{2\pi r^2} \int\limits_{\gamma_r}\, \frac{|\sigma(z) |^{m+2} }{\, \left(\mathrm{Im}\,\sigma(z)\right)^{m+1} } \, |dz| \right)^{\frac{1}{m+1}} \leqslant \left(\frac{C}{2\pi}\right)^{\frac{1}{m+1}} \end{align*} $$

for a.a. $r\in (0,1)$ .

Thus, by Lemma 2.4, we come to the estimate (3.2).

Corollary 3.2 Let $f: \mathbb {B}\to \mathbb {C}$ be a regular homeomorphic solution of equation (1.6) which belongs to Sobolev class $W_{\mathrm {loc }}^{1,2}$ , and normalized by $f(0) = 0$ and $K>0$ . Assume that the coefficient $\sigma : \mathbb {B}\to \mathbb {C}$ satisfies the following condition:

(3.3) $$ \begin{align} \frac{|\sigma (z) |^{m+2} }{\, \left(\mathrm{Im}\,\sigma(z)\right)^{m+1} }\, \leqslant K\, |z| \end{align} $$

for a.a. $z\in \mathbb {B}$ . Then

(3.4) $$ \begin{align} \limsup\limits_{z\to 0} \, \frac{|f(z)|}{|z|}\geqslant K^{-\frac{1}{m}}\,.\end{align} $$

Proof Indeed, in view of the condition (3.3), we have

$$ \begin{align*}\int\limits_{\gamma_r} \frac{|\sigma(z) |^{m+2} }{\, \left(\mathrm{Im}\,\sigma(z)\right)^{m+1} }\, |dz| \leqslant \int\limits_{\gamma_r} \, K|z|\, |dz|=2\pi K\, r^{2}\,.\end{align*} $$

Next, applying Theorem 3.1 with the parameter $C=2\pi K$ , we get the estimate (3.4).

We illustrate the last bound by the following example.

Example 3.3 Fix $k>0$ and consider the equation

(3.5) $$ \begin{align} f_\theta=\frac{i}{k^m}\,r |f_r|^m \, f_r \end{align} $$

in the unit disk $\mathbb {B}$ .

Let $f=kre^{i\theta }$ . Obviously, the mapping f belongs to the Sobolev class $W^{1,2}(\mathbb {B})$ . The partial derivatives of f with respect to $\theta $ and r are

$$ \begin{align*}f_\theta=ki\,re^{i\theta}, f_r=k\, e^{i\theta},\end{align*} $$

and $J_f(re^{i\theta })= \frac {1}{r} \, \mathrm {Im}\, \left (\overline {f_r}\, f_\theta \right )=k^2>0\,. $

Now, we show that the mapping $f=k\, re^{i\theta }$ is a solution of equation (3.5). Clearly,

$$ \begin{align*}\sigma=\frac{f_\theta}{|f_r|^m f_r}=\frac{i}{k^m}\,r \,.\end{align*} $$

Thus, (3.1) holds, since

$$ \begin{align*}\int\limits_{\gamma_r} \frac{|\sigma(z) |^{m+2} }{\, \left(\mathrm{Im}\,\sigma(z)\right)^{m+1} }\, |dz| =C \, r^2\,, \end{align*} $$

where $C=\frac {2\pi }{k^m}$ .

On the other hand, $ \lim \limits _{z\to 0} \, \frac {|f(z)|}{|z|}=k\,.$

Remark 3.4 The estimate (3.2) is sharp. It is easy to check that it is attained for a mapping $f=\left (\frac {2\pi }{C}\right )^{\frac {1}{m}}z$ .

By Theorem 2.6, similarly to the proof of Theorem 3.1, we obtain the following statement.

Theorem 3.5 Let $f\colon \mathbb {B}\to \mathbb {C}$ be a regular homeomorphic solution of equation (1.6) which belongs to Sobolev class $W^{1,2}_{\mathrm {loc}},$ and normalized by $f(0)=0.$ Suppose that

$$ \begin{align*}\sigma_{0}=\liminf\limits_{\varepsilon\rightarrow 0}\frac{1}{\pi \varepsilon^2}\int\limits_{B_{\varepsilon}} \frac{|\sigma(z)|^{m+2}}{|z| \left( \mathrm{Im}\, \sigma(z)\right)^{m+1}}\,dxdy. \end{align*} $$

1) If $\sigma _{0}\in (0,\infty )$ , then

$$ \begin{align*}\limsup\limits_{z\rightarrow 0}\frac{|f(z)|}{|z|}\geqslant c_{m}\,\sigma_{0}^{-\frac{1}{m}}, \end{align*} $$

where $c_{m}$ is a positive constant depending on the parameter $m.$

2) If $\sigma _{0}=0$ , then

$$ \begin{align*}\limsup\limits_{z\rightarrow 0}\frac{|f(z)|}{|z|}=\infty. \end{align*} $$

Example 3.6 Let $k>0$ and $\alpha \in (1,m+1)$ . Consider the equation

(3.6) $$ \begin{align} f_{\theta}=i k r^{\alpha}|f_{r}|^{m}f_{r} \end{align} $$

in the unit disk $\mathbb {B}.$

The mapping

$$ \begin{align*}f=k^{-\frac{1}{m}}\beta^{\frac{m+1}{m}}r^{\frac{m+1-\alpha}{m}} e^{i\theta}\,, \quad \beta=\frac{m}{m+1-\alpha}\,,\end{align*} $$

belongs to the Sobolev class $W^{1,2}_{\mathrm {loc}}(\mathbb {B})$ . Its partial derivatives with respect to r and $\theta $ are

$$ \begin{align*}f_{\theta}=ik^{-\frac{1}{m}}\beta^{\frac{m+1}{m}}r^{\frac{m+1-\alpha}{m}} e^{i\theta},\quad f_{r}=k^{-\frac{1}{m}}\beta^{\frac{1}{m}}r^{\frac{1-\alpha}{m}} e^{i\theta}\,.\end{align*} $$

It is easy to see that the mapping $f=k^{-\frac {1}{m}}\beta ^{\frac {m+1}{m}}r^{\frac {m+1-\alpha }{m}} e^{i\theta }$ is a regular homeomorphic solution of equation (3.6). Clearly,

$$ \begin{align*}\sigma=\frac{f_{\theta}}{|f_{r}|^{m}f_{r}}=i k r^{\alpha}.\end{align*} $$

The condition $\sigma _0=0$ in Theorem 3.5 is fulfilled, since

$$ \begin{align*}\lim\limits_{\varepsilon\rightarrow 0}\frac{1}{\pi \varepsilon^2}\int\limits_{B_{\varepsilon}} \frac{|\sigma(z)|^{m+2}}{|z| \left( \mathrm{Im}\, \sigma(z)\right)^{m+1}}\,dxdy =0.\end{align*} $$

By a direct calculation, $|f(z)|/|z|\to \infty $ as $z\to 0$ .