On a Schoenberg-type conjecture
Given an arbitrary polynomial \(P_n(z)= z^n-a_1z^{n-1} +\cdots + (-1)^na_n= \prod^n_{j=1} (z-z_j)\) satisfying the condition \(\sum^n_{j=1} z_j=a_1 =0\). Put \(R(P_n): =({1\over n}\sum^n_{j=1} |z_j|^2)^{1/2}\) and \(S(P_n): =({1\over n}\sum^n_{j=1} |z_j|^4)^{1/4}\). The Schoenberg conjecture states: (*) \(R(P_n') \leq\sqrt {{n-2\over n-1}} R(P_n)\), with equality if and only if all the zeros \(z_j\) of \(P_n(z)\) lie on straight lines. Authors propose the following conjecture: (**) \({n-4\over n-1} S(P_n)^4+ {2\over n-1}R (P_n)^4 \geq S(P_n')^4\), with equality if and only if all the zeros \(z_j\) of \(P_n(z)\) lie on straight lines through the origin in \(\mathbb{C}\). In this paper the relation (**) has been proved by direct calculation for some special form of \(P_n(z)\) (e.g. \(P_n (z)= (z^k-a^k)^{m_1} (z^k-b^k)^{m_2}\), \(a,b\in \mathbb{C}\), \(k\geq 2\), \(m_1, m_2 \geq 1)\).
scientific article; zbMATH DE number 1334864