Keywords: polynomial, zeros, EneströmKakeya theorem
American Journal of Mathematical Analysis, 2014 2 (1),
pp 1518.
DOI: 10.12691/ajma214
Received November 03, 2013; Revised March 03, 2014; Accepted March 11, 2014
Copyright: © 2014 Science and Education Publishing. All Rights Reserved.
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
 (1) 
The EneströmKakeya 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 discretetime 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 EneströmKakeya 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
 (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
 (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
 (1) 
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.
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