Journal of Inequalities and Applications 
Volume 4 (1999), Issue 3, Pages 183-213
doi:10.1155/S1025583499000363

A conjecture of Schoenberg

M. G. de Bruin,1 K. G. Ivanov,2 and A. Sharma3

 1Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics, Delft University of Technology, P.O. Box 5031, Delft 2600 GA, The Netherlands
2lnstitute of Mathematics, Bulgarian Academy of Sciences, P.O. Box 373, Sofia 1090, Bulgaria
3Department of Mathematical Sciences, University of Alberta, Alberta, Edmonton T6G 2G1, Canada
Received 16 October 1998; Revised 12 November 1998

Abstract

For an arbitrary polynomial Pn(z)=∏1n(z−zj) with the sum of all zeros equal to zero, ∑1nzj=0, the quadratic mean radius is defined by R(Pn):=(1n∑1n|zj|2)1/2. Schoenberg conjectured that the quadratic mean radii of Pn and Pn satisfy R(P′n)≤n−2n−1R(Pn),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 n=3 and for polynomials of the form zn+akzn−k.

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.