Geodesic
A generalization of the Gauss-Lucas theorem
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 481-486 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 12D10, 26C05, 30C15
Keywords: polynomials; location of zeros; convex hull of the zeros; Gauss-Lucas theorem 
@article{CMJ_2008_58_2_a12,
     author = {D{\'\i}az-Barrero, J. L. and Egozcue, J. J.},
     title = {A generalization of the {Gauss-Lucas} theorem},
     journal = {Czechoslovak Mathematical Journal},
     pages = {481--486},
     year = {2008},
     volume = {58},
     number = {2},
     mrnumber = {2411103},
     zbl = {1174.12001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a12/}
}
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Díaz-Barrero, J. L.; Egozcue, J. J. A generalization of the Gauss-Lucas theorem. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 481-486. http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a12/