  Let $u$ and $v$ be harmonic in $ \Omega \subset \mathbb{R}^n$ functions with
the same zero set $Z$. We show that the ratio $f$ of such functions is always
well-defined and is real analytic. Moreover it satisfies the maximum and
minimum principles. For $n=3$ we also prove the Harnack inequality and the
gradient estimate for the ratios of harmonic functions, namely ${
\sup\limits_{K} |f| \leq C \inf\limits_{K}| f| \quad \& \quad \sup\limits_{K}
|\nabla f| \leq C \inf\limits_{K}| f| }$ for any compact subset $K$ of
$\Omega$, where the constant $C$ depends on $K$, $Z$, $\Omega$ only. In
dimension two the first inequality follows from the boundary Harnack principle
and the second from the gradient estimate recently obtained by Mangoubi. It is
an open question whether these inequalities remain true in higher dimensions
($n \geq 4$).
