Ratio of measure of level region for harmonic functions

Question

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 $$ c\leq \frac{m(B_1(0)\cap\{u(x)>0\} )}{m(B_1(0)\cap\{u(x)<0\} )}\leq c^{-1} $$

Answer 1

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