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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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