On the massiveness of exceptional sets of the maximum modulus principle

We consider the sets $E_{\nu}(f)=\{z\colon |f(z)|\geqslant \nu\}$ for $\nu>\nu_0(f):=\limsup_{z\to\partial D}|f(z)|$ in the disc $D=\{z\colon |z|<1\}$, where $f(z)$, $z=x+iy$, are complex-valued functions defined on $D$ and having certain smoothness properties with respect to the real variables $x$ and $y$. We obtain estimates for some metric properties of the sets $E_{\nu}(f)$. For example, we prove that, if $\Delta f\in L_1(D)$, then the hyperbolic area of the set $E_\nu(f)$ cannot grow more rapidly than $\nu^{-1-o(1)}$ as $\nu\to 0$, where $o(1)$ is positive, and, if $f_{\bar{z}}\in L_2(D)$, then this area cannot grow more rapidly than $\nu^{-2-o(1)}$. The orders of these estimates with respect to $\nu$ are sharp.