Polar Convexity and Critical Points of Polynomials

Blagovest Sendov; Hristo Sendov; Chun Wang

doi:10.68381/jca26034

Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

Zeros and critical points of polynomials
Gauss-Lucas' Theorem
Laguerre's Theorem
polar derivative
pole of a set
polar convexity

Mathematics Subject Classification

30C10

References

G. Pólya, G. Szegö: Problems and Theorems in Analysis II, Springer, Berlin et al. (1976).https://doi.org/10.1007/978-1-4757-6292-1 Google Scholar
Q. I. Rahman, G. Schmeisser: Analytic Theory of Polynomials, Oxford University Press, New York et al. (2002).https://doi.org/10.1093/oso/9780198534938.001.0001 Google Scholar
R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.

[1] G. Pólya, G. Szegö: Problems and Theorems in Analysis II, Springer, Berlin et al. (1976).https://doi.org/10.1007/978-1-4757-6292-1 Google Scholar

[2] Q. I. Rahman, G. Schmeisser: Analytic Theory of Polynomials, Oxford University Press, New York et al. (2002).https://doi.org/10.1093/oso/9780198534938.001.0001 Google Scholar

[3] R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173 Google Scholar