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Bounds for the zeros of polynomials from matrix inequalities. (English) Zbl 1037.30006
Summary: We apply several matrix inequalities of the Frobenius companion matrices of monic polynomials to derive new bounds for the zeros of the polynomials. Our analysis enables us to improve an earlier bound of A. A. Abdurakhmanov [Math. USSR, Sb. 59, 39–51 (1988); translation from Mat. Sb., Nov. Ser. 131(173), No. 1(9), 40–51 (Russian) (1986; Zbl 0621.47007)] and to give a different proof of a known bound of M. Fujii and F. Kubo [Proc. Am. Math. Soc. 117, No. 2, 359–361 (1993; Zbl 0781.47013)].

MSC:
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
26C10 Real polynomials: location of zeros
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
65H05 Numerical computation of solutions to single equations
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