Abstract
Given a set of points in the complex plane, an incomplete polynomial is defined as the one which has these points as zeros except one of them. The classical result known as Gauss-Lucas theorem on the location of zeros of polynomials and their derivatives is extended to convex linear combinations of incomplete polynomials. An integral representation of convex linear combinations of incomplete polynomials is also given.
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References
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Díaz-Barrero, J.L., Egozcue, J.J. A generalization of the Gauss-Lucas theorem. Czech Math J 58, 481–486 (2008). https://doi.org/10.1007/s10587-008-0029-4
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DOI: https://doi.org/10.1007/s10587-008-0029-4