A Rolle Type Theorem for Cyclicity of Zeros of Families of Analytic Functions |
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Authors: | Email author" target="_blank">Alexander?BrudnyiEmail author |
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Institution: | 1.Department of Mathematics and Statistics,University of Calgary,Calgary,Canada |
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Abstract: | Let \({\{ {f_{\lambda ;j}}\} _{\lambda \in V;1 \leqslant j \leqslant k}}\) be families of holomorphic functions in the open unit disk \({\text{D}} \subset {\Bbb C}\) ? ? depending holomorphically on a parameter λ ∈ V ? ? 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 ∈ ?, 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 . |
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