Journal of Convex Analysis 27 (2020), No. 3, 791--810
Copyright Heldermann Verlag 2020
Sets in the Complex Plane Mapped into Convex Ones by Möbius Transformations
Inst. of Information and Communication Technologies, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
Dept. of Statistical and Actuarial Sciences, Western University, London, Ont. N6A 5B7, Canada
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.
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 non-degenerate 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, piece-wise smooth, regular boundary.
Keywords: 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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