ag.algebraic geometry – Order of vanishing of an analytic function along its vanishing locus

Suppose $F: {mathbb C}^n to {mathbb C}^k$ is a holomorphic/analytic function with vanishing locus $V_F = F^{-1}(0)$. Can one prove that there are positive real numbers $C>0$ and $delta>0$ such that within a neighborhood of $V_F$ there holds
$$|F(x)| geq C d(x, V_F)^delta$$
(or at least inside a compact region)? Here $d$ is the distance function of the Euclidean (or any) metric.