A Rolle Type Theorem for Cyclicity of Zeros of Families of Analytic Functions

Abstract

Let $${\{ {f_{\lambda ;j}}\} _{\lambda \in V;1 \leqslant j \leqslant k}}$$ be families of holomorphic functions in the open unit disk $${\text{D}} \subset {\Bbb C}$$ ⊂ ℂ depending holomorphically on a parameter λ ∈ V ⊂ ℂ n . We establish a Rolle type theorem for the generalized multiplicity (called cyclicity) of zeros of the family of univariate holomorphic functions $${\left\{ {\sum\nolimits_{j = 1}^k {{f_{\lambda ;j}}} } \right\}_{\lambda \in V}}$$ at 0 ∈ D. As a corollary, we estimate the cyclicity of the family of generalized exponential polynomials, that is, the family of entire functions of the form $$\sum\nolimits_{k = 1}^m {{P_k}(z){e^{{Q_k}(z)}}} $$ , z ∈ ℂ, 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 .