×

Some generalizations of the Eneström-Kakeya theorem. (English) Zbl 1350.30009

The main result of the paper is the following.
Theorem. Let \(p(z) =\sum\limits_{j=0}^n a_jz^j\) be a polynomial of degree \(n\) with real coefficients. If for some real numbers \(\alpha\) and \(\beta\) \[ a_0-\beta \leq a_1\leq a_2 \leq \dots \leq a_n +\alpha, \] then all the zeros of \(p(z)\) lie in the disc \[ \left| z+\frac{\alpha}{a_n} \right|\leq \frac{1}{|a_n|}\big[ a_n +\alpha -a_0 + \beta +|\beta | + |a_0|\big]. \] Also an analogous theorem is proved, where, for the coefficients, \[ a_0 - s \leq a_1 \leq a_2 \leq \dots \leq a_{\lambda -1}\leq a_{\lambda} \geq a_{\lambda +1} \geq \dots \geq a_{n-1}\geq a_n +t, \] holds with some real numbers \(t, s\) and for some integer \(\lambda\), with \(0< \lambda < n\).
The results extend and generalize the result of A. Aziz and B. A. Zargar [Anal. Theory Appl. 28, No. 2, 180–188 (2012; Zbl 1265.30011)].

MSC:

30C10 Polynomials and rational functions of one complex variable
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)

Citations:

Zbl 1265.30011

Software:

Matlab
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] N. Anderson, E. B. Saff, and R. S. Varga, On the Enestr”om-Kakeya theorem and its sharpness, Linear Algebra Appl. 28 (1979), 5–16. · Zbl 0423.15007
[2] N. Anderson, E. B. Saff, and R. S. Varga, An extension of the Enestr”om-Kakeya theorem and its sharpness, SIAM J. Math. Anal. 12 (1981), 10–22. · Zbl 0455.30006
[3] A. Aziz and B. A. Zargar, Some extensions of Enestr”om-Kakeya theorem, Glas. Mat. Ser. III 31 (51) (1996), 239–244. · Zbl 0867.30004
[4] A. Aziz and B. A. Zargar, Bounds for the zeros of a polynomial with restricted coefficients, Appl. Math. (Irvine) 3 (2012), 30–33.
[5] G. Enestr”om, H”arledning af en allm”an formel f”or antalet pension”arer, som vid en godtycklig tidpunkt f”orefinnas inom en sluten pensionskassa, Stockh. ”Ofv. L. 6 (1893), 405–415. (Swedish) · JFM 25.0360.01
[6] A. Joyal, G. Labelle, and Q. I. Rahman, On the location of zeros of a polynomial, Canad. Math. Bull. 10 (1967), 55–63. · Zbl 0152.06102
[7] S. Kakeya, On the limits of the roots of an algebraic equation with positive coefficients, Tohoku Math. J. 2 (1912), 140–142. · JFM 43.0147.03
[8] M. Kovacevi’c and I. Milovanovi’c, On a generalization of the Enestr”om-Kakeya theorem, Pure Math. Appl. Ser. A 3 (1992), 43–47.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.