Order of vanishing of an analytic function along its vanishing locus

Question

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.