On zeros of components of analytic curves and of derivatives of analytic functions

For an analytic curve $F=(f_1, f_2, \dots, f_m), m\ge 1,$ in a domain $G$, let $d_F(z)$ be the radius of the largest disk centered at $z\in G$ in which any $f_j$ does not vanish. Lower estimates for $\sup\{d_F(z)\colon z\in G\}$ and $\sup\{d_F(z)l(|z|)\colon z\in G\}$ ($l$ be a positive continuous increasing to $+\infty$ function) are given. The cases $G=\mathbb C$ and $G=\{z\colon |z|<1\}$ are investigated.