Ratios of harmonic functions with the same zero set

We study the ratio of harmonic functions $u,v$, which have the same zero set $Z$ in the unit ball $B\subset {\mathbb{R}}^n$. The ratio $f=u\mathrm{/}v$ can be extended to a real analytic nowhere vanishing function in $B$. We prove the Harnack inequality and the gradient estimate for such ratios in any dimension: for a given compact set $K\subset B$ we show that ${\sup }_K\mathrm{\mid }f\mathrm{\mid }\le C_1{\inf }_K\mathrm{\mid }f\mathrm{\mid }$ and ${\sup }_K\mid \mathrm{\nabla }f\mid \le C_2{\inf }_K\mathrm{\mid }f\mathrm{\mid }$, where $C_1$ and $C_2$ depend on $K$ and $Z$ only. In dimension two we specify the dependence of the constants on $Z$ in these inequalities by showing that only the number of nodal domains of $u$, i.e. the number of connected components of $B\setminus Z$, plays a role.