Geodesic
Sets in the Complex Plane Mapped into Convex Ones by Möbius Transformations
Journal of convex analysis, Tome 27 (2020) no. 3, pp. 791-81
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A set A in the extended complex plane is called convex with respect to a pole u, if for any two points z1 and z2 from the set, the arc from z1 to z2 on the unique circle through u, z1, and z2, opposite of u is contained in A. In that case we say that u is a pole of A. When u = ∞, this notion coincides with the usual convexity. Polar convexity, allows one to extend and/or strengthen several classical results about the location of the critical points of polynomials, such as the Gauss-Lucas' and the Laguerre's theorem.
Classification : 30C10
Mots-clés : Zeros and critical points of polynomials, Gauss-Lucas' Theorem, Laguerre's Theorem, polar derivative, pole of a set, polar convexity, osculating circle 
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     author = {B. Sendov and H. Sendov},
     title = {Sets in the {Complex} {Plane} {Mapped} into {Convex} {Ones} by {M\"obius} {Transformations}},
     journal = {Journal of convex analysis},
     pages = {791--81},
     year = {2020},
     volume = {27},
     number = {3},
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B. Sendov; H. Sendov. Sets in the Complex Plane Mapped into Convex Ones by Möbius Transformations. Journal of convex analysis, Tome 27 (2020) no. 3, pp. 791-81. http://geodesic.mathdoc.fr/item/JCA_2020_27_3_JCA_2020_27_3_a0/