A conjecture of Schoenberg

Abstract

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