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Journal of Mathematical Analysis and Applications
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On the Zeros of Polynomials with Concentration at Low Degrees, II

On the zeros of polynomials with concentration at low degrees. II
Authors: Beauzamy, B.; Chou, S.;

On the Zeros of Polynomials with Concentration at Low Degrees, II

Abstract

[For part I see the second author in ibid. 149, No. 2, 424-436 (1990; Zbl 0714.30008).] A complex polynomial \(P(z)=\sum^ n_{j=0} a_ jz^ j\) is defined to have concentration \(d(0k}z^{-1}_ j \right | \leq C(d,k):=(9^{k+3}/d^ 3) (2(1+d)/d)^ k \text{ and } \prod_{j>k} | z_ j | \geq 1/C(d,k) \text{ hold}. \] The proof is incomplete, since in its final step the inequality \(C(d/2,k-1) \leq C(d,k)\) is used, which does not hold for large \(k\) and fixed \(d\). Among further results a circle, centered at the origin, is constructed, that contains at least one zero of \(P\) and the following theorem is proved. If \(P\) is a Hurwitz polynomial (i.e. all \(a_ j>0\) and all \(\text{Re} z_ j \leq 0)\) with concentration \(d\) at degree \(k\), then \[ \sum^ n_{j=1} (1-z_ j)^{-1} \leq C_ H(d,k):=9 \log(1/d)+(11k+9) \log 2 \] holds, which also yields an upper bound for \(| \sum_{j>k} z_ j^{-1} |\) and a lower bound for \(\prod_{j>k} | z_ j |\). Furthermore the generalized Bernstein inequality \(\| P' \|\leq C_ H (d,k) \| P \|\) is obtained, where \(\| P \|:=\max_{| z |=1} | P(z) |\). The results on Hurwitz polynomials are extended to special classes of entire functions of order \(\leq 1\).

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Keywords

polynomials with concentration at low degree, Hurwitz polynomial, Bernstein inequality, Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral), bounds of zeros, Applied Mathematics, Inequalities in the complex plane, Analysis

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
1
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