Inequalities for central moments and spreads of matrices (Q781724)

The authors prove several inequalities involving the first four central moments of a distribution. In the case of \[ \bar{x}=\frac{1}{n}\sum_{i=1}^nx_i,\quad m_r=\frac{1}{n}\sum_{i=1}^n(x_i-\bar{x})^r, \quad r=1,2,3,4, \] they present two applications. First, let \(A\in\mathbb{C}^{n\times n}\) have real eigenvalues \(\lambda_1\ge\ldots\ge\lambda_n\). The authors find bounds for \(\lambda_1\), \(\lambda_n\), and \(\lambda_1-\lambda_n\). For example (Theorem 7), \[ \lambda_n\le\frac{\mathrm{tr}\,A}{n}-\Big[\frac{n^2-3n+3}{n^3(n-1)^3}\Big]^\frac{1}{4} \frac{\mathrm{tr}\,B^2}{(\mathrm{tr}\,B^4)^\frac{1}{4}} \] and \[ \lambda_1\ge\frac{\mathrm{tr}\,A}{n}+\Big[\frac{n^2-3n+3}{n^3(n-1)^3}\Big]^\frac{1}{4} \frac{\mathrm{tr}\,B^2}{(\mathrm{tr}\,B^4)^\frac{1}{4}}, \] where \[ B=A-\frac{\mathrm{tr}\,A}{n}I. \] Second, consider a polynomial \[ f(x)=x^n+a_2x^{n-2}+\dots+a_{n-1}x+a_n,\quad n\ge 5, \] with real coefficients and real zeros \(x_1\ge\dots\ge x_n\). (Note that \(a_1=0\).) The authors find bounds for \(x_1\), \(x_n\), and \(x_1-x_n\). For example (Theorem 10), \[ x_n\le-\Big[\frac{8(n^2-3n+3)}{n^3(n-1)^3}\frac{a_2^4}{a_2^2-2a_4}\Big]^\frac{1}{4} \] and \[ x_1\ge\Big[\frac{8(n^2-3n+3)}{n^3(n-1)^3}\frac{a_2^4}{a_2^2-2a_4}\Big]^\frac{1}{4}. \]