Let $u$ be a harmonic function defined on $B\_1(0)\subset\mathbb{R}^2$, $u(0)=0$, and $\{x\in B\_1(0):u(x)>0\}$ is simply connected. Is there a universal constant $c>0$ satisfying that
$$
c\leq \frac{m(B\_1(0)\cap\{u(x)>0\} )}{m(B\_1(0)\cap\{u(x)<0\} )}\leq c^{-1}
$$



Ratio of measure of level region for harmonic functions
The answer is negative. To construct a counterexample, use an entire function $f$ which satisfies $|f(z)|<1$ in ${\mathbf{C}}\backslash D$, where $D$ is a half-strip $\{ x+iy:x>0,|y|<\pi \}$. Then $v={\mathrm{Re}}f-1$ is harmonic, and negative in ${\mathbf{C}}\backslash D$. Let $D\_1=\{ z: u(z)>0\}\subset D$, and
$z\_1\in\partial D\_1$. Then $v\_1(z)=v(z-1)$ satisfies $v\_1(0)=0$ and for sufficiently large $R$, $u(z)=v\_1(Rz)$ will be positive on a subset of the unit disk of arbitrarily small area.


A function $f$ with required property can be constructed as a Cauchy integral
$$f(z)=\int\_{\partial D}\frac{e^{e^\zeta}d\zeta}{\zeta-z},$$
for details about this construction, see, for example
W. Hayman, Meromorphic functions, Chap. 4, 4.1.
See also [Entire function bounded at every line](https://mathoverflow.net/questions/190837/entire-function-bounded-at-every-line/190874#190874)


