A Rolle type theorem for cyclicity of zeros of families of analytic functions

Let $\{f_{\lambda; j}\}_{\lambda\in V; 1\le j\le k}$ be families of holomorphic functions in the open unit disk $\Di\subset\Co$ depending holomorphically on a parameter $\lambda\in V\subset \Co^n$. We establish a Rolle type theorem for the generalized multiplicity (called {\em cyclicity}) of zero of the family of univariate holomorphic functions ${\sum_{j=1}^k f_{\lambda;j}}_{\lambda\in V}$ at $0\in\Di$. As a corollary, we estimate the cyclicity of the family of generalized exponential polynomials, that is, the family of entire functions of the form $\sum_{k=1}^m P_k(z)e^{Q_k(z)}$, $z\in\Co$, where $P_k$ and $Q_k$ are holomorphic polynomials of degrees $p$ and $q$, respectively, parameterized by vectors of coefficients of $P_k$ and $Q_k$.