Journal of Convex Analysis 26 (2019), No. 2, 635--660
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 Gauss-Lucas' 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, Gauss-Lucas' Theorem, Laguerre's Theorem, polar derivative, pole of a set, polar convexity.
MSC: 30C10
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