Is the following true?

**Conjecture?** *Let $U \subset \mathbb{C}^n$ be open and $\eta : U \to \mathbb{C}$ be holomorphic. Denote by $B(z,r)$ the usual ball of radius $r$. There is a constant $\kappa<\infty$ such that, for all $B(z,2r) \subset U$, if $\eta$ is nonzero on $B(z,2r)$ then $$\sup\_{z \in B(z,r)} |\eta(z)| \leq \kappa \inf\_{z \in B(z,r)} |\eta(z)|.$$*

By way of the Weierstrass preparation theorem, it might suffice to show a related estimate for polynomials of a fixed degree $m$. It may also be necessary to assume that $\eta$ is in fact holomorphic on some $V$ with $U \subset \subset V$.