Graphs of zeros of analytic families

Let $\mathcal{F}:=\{f_{\lambda}\}$ be a family of holomorphic functions in a domain $D\subset\mathbb{C}$ depending holomorphically on $\lambda\in U\subset\mathbb{C}^{n}$. We study the distribution of zeros of $\{f_{\lambda}\}$ in a subdomain $R\subset\subset D$ whose boundary is a closed non-singular analytic curve. As an application, we obtain several results about distributions of zeros of families of generalized exponential polynomials and displacement maps related to certain ODE's.