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.
