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An analog of the Poincaré separation theorem for normal matrices and the Gauss-Lucas theorem. (English. Russian original) Zbl 1051.15013
Funct. Anal. Appl. 37, No. 3, 232-235 (2003); translation from Funkts. Anal. Prilozh. 37, No. 3, 85-88 (2003).
The author states a number of theorems on inequalities of which the following are typical (proofs appear elsewhere). Let \(q(\lambda)\) be a monic polynomial of degree \(m\) over \(\mathbb{C}\), and denote the roots of \(q(\lambda)\) by \(\lambda_{1},\dots,\lambda_{m}\) and the roots of its derived polynomial \(q^{\prime}(\lambda)\) by \(\mu_{1},\dots,\mu_{m-1}\).
(1) If \(q_{1}(\lambda)\) is another monic polynomial of degree \(m-1\). Then there exists a normal \(m\times m\) matrix \(A\) such that \(A\) has characteristic polynomial \(q(\lambda)\) and its principal minor \(A_{m-1}\) of size \((m-1)\times(m-1)\) has characteristic polynomial \(q_{1}(\lambda)\) if and only if \(q_{1}(\lambda_{k})\) lies in the convex hull of \(0\) and \(q^{\prime} (\lambda_{k})\) for each \(\lambda_{k}\).
This can be recognized as a generalization of the Cauchy-Poincaré interlacing theorem for Hermitian matrices.
(2) For all convex functions \(f:\mathbb{C}\to\mathbb{R}\), all \(\alpha\in\mathbb{C}\) and all positive integers \(p\leq m-1\) we have: \[ \binom{m-1}{p}^{-1}\sum f \left(\prod_{s=1}^{p}(\mu_{i_{s}}-\alpha)\right)\leq \binom{m}{p}^{-1}\sum f\left(\prod_{s=1}^{p}(\lambda_{i_{s}}-\alpha)\right) \] where the first sum is over all indices \(1\leq i_{1}<\dots<i_{p}\leq m-1\) and the second is over all indices \(1\leq i_{1}<\dots<i_{p}\leq m\).
This is a generalization of an inequality conjectured by N. G. de Bruijn and T. A. Springer [Proc. Akad. Wet. Amsterdam 50, 458–464 (1947; Zbl 0029.19801)].
(3) If \(\lambda_{1}+\dots+\lambda_{m}=0\), then \(m\sum_{j=1}^{m-1}| \mu _{j}| ^{2}\leq(m-2)\sum_{j=1}^{m}| \lambda_{j}| ^{2}\); and equality holds if and only if all the roots \(\lambda_{k}\) lie on the same line.
This verifies a conjecture of [I. Schoenberg, Am. Math. Mon. 93, 8–13 (1986; Zbl 0627.30001)].
Editorial remark: Other proofs of the aforementioned conjectures have appeared by R. Pereira [J. Math. Anal. Appl. 285, 336–348 (2003; Zbl 1046.47002)].

MSC:
15A42 Inequalities involving eigenvalues and eigenvectors
26D05 Inequalities for trigonometric functions and polynomials
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