A Rolle Type Theorem for Cyclicity of Zeros of Families of Analytic Functions
Tóm tắt
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
.
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