On a Schoenberg-type conjecture

For an arbitrary polynomial Pn(z)=zn−a1zn−1+a2zn−2+⋯+(−1)nan=∏1n(z−zj) with the sum of all zeros equal to zero, a1=∑1nzj=0, the quadratic mean radius is defined byR(Pn)≔1n∑1n|zj|21/2,and the quartic mean radius byS(Pn)≔1n∑1n|zj|41/4.This paper studies a Schoenberg-type conjecture using the quartic mean radius in the following form:n−4n−1S(Pn)4+2n−1R(Pn)4⩾S(Pn′)4,with equality if and only if the zeros all lie on a straight line through the origin in the complex plane.