
Journal of Convex Analysis 26 (2019), No. 2, 635660 Copyright Heldermann Verlag 2019 Polar Convexity and Critical Points of Polynomials Blagovest Sendov Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. / bl. 25A, 1113 Sofia, Bulgaria acad@sendov.com Hristo Sendov Department of Statistical and Actuarial Sciences, Western University, 1151 Richmond Str., London, Ont., Canada N6A 5B7 hssendov@stats.uwo.ca Chun Wang Department of Statistical and Actuarial Sciences, Western University, 1151 Richmond Str., London, Ont., Canada N6A 5B7 cwang593@uwo.ca 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 GaussLucas' 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. Keywords: Zeros and critical points of polynomials, GaussLucas' Theorem, Laguerre's Theorem, polar derivative, pole of a set, polar convexity. MSC: 30C10 [ Fulltextpdf (583 KB)] for subscribers only. 