Geodesic
Polar Convexity and Critical Points of Polynomials
Journal of convex analysis, Tome 26 (2019) no. 2, pp. 635-66
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A set A, in the extended complex plane, is called convex with respect to a pole u, if for any x,y in A the arc on the unique circle through x,y, and u, that connects x and y but does not contain u, is in A. If the pole u is taken at infinity, this notion reduces to the usual convexity. Polar convexity is connected with the classical Gauss-Lucas' and Laguerre's theorems for complex polynomials. If a set is convex with respect to u and contains the zeros of a polynomial, then it contains the zeros of its polar derivative with respect to u. A set may be convex with respect to more than one pole. The main goal of this article is to find the relationships between a set in the extended complex plane and its poles.
Classification : 30C10
Mots-clés : Zeros and critical points of polynomials, Gauss-Lucas' Theorem, Laguerre's Theorem, polar derivative, pole of a set, polar convexity 
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     author = {B. Sendov and H. Sendov and C. Wang},
     title = {Polar {Convexity} and {Critical} {Points} of {Polynomials}},
     journal = {Journal of convex analysis},
     pages = {635--66},
     year = {2019},
     volume = {26},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JCA_2019_26_2_JCA_2019_26_2_a14/}
}
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B. Sendov; H. Sendov; C. Wang. Polar Convexity and Critical Points of Polynomials. Journal of convex analysis, Tome 26 (2019) no. 2, pp. 635-66. http://geodesic.mathdoc.fr/item/JCA_2019_26_2_JCA_2019_26_2_a14/