Videnskiĭ, I. V. Zeros of the derivative of a rational function and coinvariant subspaces for the shift operator on the Bergman space. (English. Russian original) Zbl 1069.30008 J. Math. Sci., New York 120, No. 5, 1657-1661 (2004); translation from Zap. Nauchn. Semin. POMI 282, 26-33 (2001). Summary: If all \(n\) \((n>1)\) zeros of a rational function \(r\) with simple poles are in a half-plane, then the derivative of \(r\) has at least one zero in the same half-plane. This result is used to prove that the number of zeros of a linear combination of \(n\) Bergman kernels in the unit disk may range from 0 to \(2n-3\). Cited in 2 Documents MSC: 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) 47B38 Linear operators on function spaces (general) × Cite Format Result Cite Review PDF Full Text: DOI