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Open Access Research Article

A conjecture of Schoenberg

MG de Bruin1*, KG Ivanov2 and A Sharma3

  • * Corresponding author: MG de Bruin

Author Affiliations

1 Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics, Delft University of Technology, P.O. Box 5031, Delft 2600 GA, The Netherlands

2 lnstitute of Mathematics, Bulgarian Academy of Sciences, P.O. Box 373, Sofia 1090, Bulgaria

3 Department of Mathematical Sciences, University of Alberta, Alberta, Edmonton T6G 2G1, Canada

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Journal of Inequalities and Applications 1999, 1999:838171  doi:10.1155/S1025583499000363

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/1999/3/838171


Received:16 October 1998
Revisions received:12 November 1998

© 1999 de Bruin et al.

For an arbitrary polynomial with the sum of all zeros equal to zero, , the quadratic mean radius is defined by Schoenberg conjectured that the quadratic mean radii of and satisfy where equality holds if and only if the zeros all lie on a straight line through the origin in the complex plane (this includes the simple case when all zeros are real) and proved this conjecture for and for polynomials of the form .

It is the purpose of this paper to prove the conjecture for three other classes of polynomials. One of these classes reduces for a special choice of the parameters to a previous extension due to the second and third authors.

Keywords:
Geometry of zeros; Weighted sums; Inequalities