A conjecture of Schoenberg. Issue 3 (1999) - de Bruin, M. G.; Ivanov, K. G.; Sharma, A.
Abstract : For an arbitrary polynomialP n ( z ) = ∏ 1 n ( z − z j ) with the sum of all zeros equal to zero, ∑ 1 n z j = 0, the quadratic mean radius is defined byR ( P n ) : = ( 1 n ∑ 1 n | z j | 2 ) 1 / 2 . Schoenberg conjectured that the quadratic mean radii ofP n andP n satisfyR ( P ′ n ) ≤ n − 2 n − 1 R ( P n ), 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 forn = 3 and for polynomials of the formz n + a k z n − 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.