Sets in the Complex Plane Mapped into Convex Ones by Möbius Transformations

A set A in the extended complex plane is called convex with respect to a pole u, if for any two points

and

z_2 z 2

from the set, the arc from

z_1 z 1

to

z_2 z 2

on the unique circle through u,

z_1 z 1

, and

z_2 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 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.