
Journal of Convex Analysis 27 (2020), No. 3, 791810 Copyright Heldermann Verlag 2020 Sets in the Complex Plane Mapped into Convex Ones by Möbius Transformations Blagovest Sendov Inst. of Information and Communication Technologies, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria acad@sendov.com Hristo Sendov Dept. of Statistical and Actuarial Sciences, Western University, London, Ont. N6A 5B7, Canada hssendov@stats.uwo.ca A set A in the extended complex plane is called convex with respect to a pole u, if for any two points z_{1} and z_{2} from the set, the arc from z_{1} to z_{2} on the unique circle through u, z_{1}, and z_{2}, 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 GaussLucas' and the Laguerre's theorem. Another way to characterize a pole of a set is through Möbius transformations. A point u is a pole of A if W(A) is a convex set, whenever W is a nondegenerate Möbius transformation, such that W(u) = ∞. The goal of this paper is to describe the set of all poles of a given set A with simple, piecewise smooth, regular boundary. Keywords: Zeros and critical points of polynomials, GaussLucas' Theorem, Laguerre's Theorem, polar derivative, pole of a set, polar convexity, osculating circle. MSC: 30C10. [ Fulltextpdf (755 KB)] for subscribers only. 