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On the Eneström-Kakeya Theorem

Gulshan Singh

Department of Mathematics, Jammu and Kashmir State Government of Education, India

Abstract: In this paper, we prove some extensions of the Eneström-Kakeya theorem by relaxing the hypothesis in different ways which in turn generalizes a result of Aziz and Zargar [Some extensions of Eneström-Kakeya Theorem, Glasnik Matematički, 31(1996), 239-244].

Keywords: polynomial, zeros, Eneström-Kakeya theorem

Received November 03, 2013; Revised March 03, 2014; Accepted March 11, 2014

Citation: Gulshan Singh, On the Eneström-Kakeya Theorem. American Journal of Mathematical Analysis, 2 (1). 15-18. January 2014. doi:10.12691/ajma-2-1-4

1. Introduction and Statement of Results

Finding the roots of a polynomial is a long standing classical problem [3,7]. The various results in the analytic theory of polynomials concerning the number of zeros in a region have been frequently investigated. It is an interesting area of research for engineers as well as mathematicians and many results on the same topic are available in the literature. Over last five decades, a large number of research papers, e.g, [1,4,5,6,8,9,13,14] and monographs [10,12,15] have been published. Polynomials in various forms have recently come under extensive revision because of their applications in linear control systems, signal processing, electrical networks, coding theory and several areas of physical sciences, where among others location of zeros and stability problems arise in a natural way. Existing results in the literature also show that there is need to find bounds for special polynomials, for example, those having restrictions on the coefficient and there is always need for refinement of results in this subject.

The following result is well known in the theory of the distribution of zeros of polynomials.

Theorem A. If is a polynomial of degree n such that

then does not vanish in

If we apply this result to the polynomial then it can be restated as:

Theorem B. Let be a polynomial of degree n such that

then all the zeros of lie in

[picture] | (1)

The Eneström-Kakeya theorem is a very strong tool to find the region in the complex plane containing all the zeros of a class of polynomials. It has been used to analyze overflow oscillation of discrete-time dynamical system [11], to investigate the properties of orthogonal wavelets [9], to determine the asymptotic behavior of zeros of the Daubechies filter [16], in addition for application to a model of high energy collisions [2].

In the literature, [1,4,5,6,14], there exist extensions and generalizations of Eneström- Kakeya theorem.

Joyal et al. [8] obtained the following generalization, by considering the coefficients to be real, instead of being only positive.

Theorem C. Let be a polynomial of degree n such that

then all the zeros of lie in

Aziz and Zargar [1] also relaxed the hypothesis of Eneström-Kakeya theorem in a different way and proved the following results.

Theorem D. Let be a polynomial of degree n such that for some

then all the zeros of lie in

Theorem E. Let be a polynomial of degree n such that for some

then all the zeros of lie in

In this paper, we prove more general result by relaxing the hypothesis in different ways which includes Theorem E as a special case.

Theorem 1. Let be a polynomial of degree n such that where and , are real numbers. If for some positive integers and for some real numbers

then all the zeros of lie in

[picture] | (2)

Remark 1. Theorem E is a special case of Theorem 1, if we take all coefficients of are real, and

The following Corollary immediately follows from Theorem 1.

Corollary 1. Let be a polynomial of degree n such that where and , are real numbers. If for some positive integers and for some real numbers

then all the zeros of lie in

If we choose and in Theorem 1, we have the following:

Corollary 2. Let be a polynomial of degree n such that where and , are real numbers. If for some real numbers

then all the zeros of lie in

Also, if we put and in Theorem 1, we get

Corollary 3. Let be a polynomial of degree n such that where and , are real numbers. If for some real number

then all the zeros of lie in

[picture] | (3)

On combining Theorem B and by taking all coefficients of to be real in the Corollary 2, the following is immediate:

Corollary 4. Let be a polynomial of degree n, if for some

then all the zeros of lie in the intersection of the discs represented by (1) and (3).

2. Proof of the Theorem 1

Consider the polynomial

Let so that and we have

if

This shows that if then if

Hence all the zeros of with lie

[picture] | (4)

But those zeros of whose modulus is less than or equal to 1 already satisfy the inequality (1). Since all the zeros of are also the zeros of , therefore it follows that all the zeros of lie in the circle defined by (1) and this completes the proof of Theorem 1.

References

[1]   | A. Aziz and B. A. Zargar, Some extensions of Enestrom-Kakeya Theorem, Glasnik Matematički, 31, (1996), 239-244.
[2]   | M. Brambilla, A. Giovannini and R. Ugoccioni, Maps of zeros of the grand canonical partition function in a statistical model of high energy collisions, J. Phys. G: Nucl. Part. Phys., 32((2006), 859-870.
[3]   | R. Fricke, Lehrbuch der Algebra, Friedrich Vieweg and Sohn Verlag, braunschweig, Germany, 1924.
[4]   | N. K. Govil and Q. I. Rahman, On the Eneström-Kakeya Theorem II, Tohoku Math. J. 20, (1968), 126-136.
[5]   | Gulshan Singh and W.M Shah, On the Eneström-Kakeya theorem, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104) No. 4, 2013, 505-512.
[6]   | Gulshan Singh and W. M. Shah, On the Eneström-Kakeya theorem, Applied Mathematics, 3(2010), 555-560.
[7]   | W. Heitzinger, W. I. Troch and G. Valentin, Praxis nichtlinearer Gleichungen, Carl Hanser Verlag, munchen, Wien, Germany, Austria, 1985.
[8]   | A. Joyal, G. Labelle and Q. I. Rahman, On the location of zeros of polynomials, Canad. Math. Bull. 10, (1967), 53-63.
[9]   | J. R. Karam, An application of the Kakeya-Eneström theorem, Proc. Int. Conf. Image Sci. in Syst. Technol. CISST, 04(2004), 354-358.
[10]   | M. Marden, Geometry of polynomials, IInd Ed. Math. Surveys 3, Amer. Math. Soc., Providence, R.I. (1966).
[11]   | T. Ooba, On companion systems with state saturation nonlinearity, IEEE Trans. Autom. Control, 50(2003), 1580-1584.
[12]   | Q. I. Rahman and G. Schmeisser, Analytic theory of polynomials, Oxford University Press, Oxford (2002).
[13]   | W. M. Shah and A. Liman, On the zeros of a certain class of polynomials and related analytic functions, Mathematicka Balkanicka, New Series, 19, (2005): Facs., 3-4.
[14]   | W. M. Shah, A. Liman and Shamim Ahmad Bhat, On the Eneström-Kakeya Theorem, International Journal of Mathematical Science, 7, (1-2) (2008), 111-120.
[15]   | T. Sheil-Small, Complex polynomials, Cambridge University Press, Cambridge (2002).
[16]   | J. Shen and G. Strang, Asymptotic analysis of Daubechies polynomials, Proc. Amer. Math. Soc., 124(1996), 3819-3833.