Bounds for the Zeros of a Polynomial with Restricted Coefficients ()
1. Introduction and Statement of Results
The following results which is due to Enestrom and Kakeya [1] is well known in the theory of the location of the zeros of polynomials.
THEOREM A. Let
be a polynomial of degree n, such that
, (1)
then
does not vanish in
.
In the literature [2-5] there exist some extensions and generalization of Enestrom-Kakeya Theorem. Joyal, Labelle and Rahman [6] extended this theorem to polynomials whose coefficients are monotonic but not necessarily non-negative by proving the following result:
THEOREM B. Let
![](https://www.scirp.org/html/5-7400449\8a8792e4-2be0-4196-b8d5-19535c6fd5c0.jpg)
then the polynomial
![](https://www.scirp.org/html/5-7400449\78130661-68cd-4021-a2b3-62d9a8af445c.jpg)
of degree n has all its zeros in
(2)
Recently Aziz and Zarger [7] relaxed the hypothesis in several ways and among other things proved the following results:
THEOREM C. Let
![](https://www.scirp.org/html/5-7400449\395d89d8-e964-4364-9030-81f7e72b0bda.jpg)
be the polynomial of degree n, such that for some k ≥ 1,
(3)
then P(z) has all its zeros in
(4)
The aim of this paper is to prove some extensions of Enestrom-Kakeya Theorem (Theorem-A) by relaxing the hypothesis in various ways. Here we shall first prove the following generalization of Theorem C which is an interesting extension of Theorem A.
2. Main Results
THEOREM 1.1. Let
be a polynomial of degree n. If for some positive numbers k and
with k ≥ 1, and ![](https://www.scirp.org/html/5-7400449\09b7e561-ccea-4180-a65b-3b72c93fd50a.jpg)
(5)
then all the zeros of P(z) lie in the closed disk
(6)
If we take
in Theorem 1.1 we obtain the following result which is a generalization of Corollary 2 ([7]).
COROLLARY 1. Let
be a polynomials of degree n. If for some positive real number![](https://www.scirp.org/html/5-7400449\3e82d5ac-d7ad-428b-b36b-4704d3100e39.jpg)
(7)
then all zeros of P(z) lie in
![](https://www.scirp.org/html/5-7400449\5ca1db8f-20cb-4efc-8a02-9e54a3a45158.jpg)
REMARK 1. Theorem 1.1 is applicable to situations when Enestrom-Kakeya Theorem gives no information. To see this consider the polynomial.
![](https://www.scirp.org/html/5-7400449\fdd7dacf-a726-4169-bc13-f5b5f160b417.jpg)
with
is a positive real number. Here EnestromKakeya Theorem is not applicable to P(z) where as Theorem 1.1 is applicable with
and according to our result, all the zeros of P(z) lie in the disk.
![](https://www.scirp.org/html/5-7400449\9069caa9-51de-45a6-9810-899c0db209e6.jpg)
which is considerably better than the bound obtained by a classical result of Caushy ([4]) which states that all the zeros of P(z) lie in
![](https://www.scirp.org/html/5-7400449\f7078a5e-3ed9-484b-a3cc-df0bbbfd226d.jpg)
where
![](https://www.scirp.org/html/5-7400449\5330ad7a-1920-4d52-82cf-875b4072f755.jpg)
Next, we present the following generalization of corollary 1 which includes Theorem 4 of [6] as a special case and considerably improves the bound obtained by Dewan and Bidkham ([8], Theorem1) for t = 0 and
.
THEOREM 1.2. Let
be a polynomial of degree n. If for some positive number
and for some non-negative integer ![](https://www.scirp.org/html/5-7400449\7de92eb7-fb0e-4205-933a-08096d5645a8.jpg)
(8)
then all the zeros of P(z) lie in
(9)
Applying Theorem 1.2 to P(tz), we get the following result:
COROLLARY 2. Let
be a polynomial of degree n. If for some positive numbers t and
with
,
![](https://www.scirp.org/html/5-7400449\113e7974-580c-471e-8edf-084ccf70d22e.jpg)
where
is a non negative integer then all the zeros of P(z), lie in
![](https://www.scirp.org/html/5-7400449\3a732457-0483-4aa4-bc45-c7f3967fa2ec.jpg)
If we assume a0 > 0, in Theorem 1.2, we obtain.
COROLLARY 3. Let
be a polynomial of degree n. If for some positive numbers
and for same non-negative integer
![](https://www.scirp.org/html/5-7400449\677a0c11-3610-413b-923d-f520a327c6e3.jpg)
![](https://www.scirp.org/html/5-7400449\77070f02-2fbf-42f7-bb22-62c2867c5eec.jpg)
then all the zeros of P(z) lie in,
(10)
Finally we present all following generalization of Theorem B due to Joyal, Labelle and Rahman which includes Theorem A as a special case.
THEOREM 1.3. Let
be a polynomial of degree n, It for some positive number
and for some non-negative integer ![](https://www.scirp.org/html/5-7400449\00f70813-8ac9-4a50-a880-0c7d73ad83fb.jpg)
![](https://www.scirp.org/html/5-7400449\25e165bc-87ea-4987-b2f5-509a2345b51b.jpg)
then all the zeros of P(z) lie in
(11)
REMARK 2. For
, Theorem 1.3 reduces to Theorem B.
3. Proofs of the Theorems
PROOF OF THEOREM 1.1. Consider
![](https://www.scirp.org/html/5-7400449\9c9ad254-337b-4521-b79e-af69caa62869.jpg)
then for
we have
![](https://www.scirp.org/html/5-7400449\f22cdf27-e1db-4ca3-bb73-06741d386fc7.jpg)
this shows that if
then
, if
![](https://www.scirp.org/html/5-7400449\f67a7ee2-0d0f-43d6-af17-9d3978a9e1e9.jpg)
therefore all the zeros of F(z), whose modulus is greater than 1 lie in the closed disk
![](https://www.scirp.org/html/5-7400449\f5876bf0-70f6-4226-b47c-475c96715b82.jpg)
But those zeros of F(z) whose modules is less than or equal to 1 already satisfy the Inequality (6).
Since all the zeros of P(z) are also the zeros of F(z). therefore it follows that all the zeros of P(z) lie in the circle defined by (6). Which completes the proof of Theorem 1.1.
PROOF OF THEOREM 1.2. Consider
![](https://www.scirp.org/html/5-7400449\39765063-6f35-4bd8-97b3-4a57746bd475.jpg)
therefore, for
and
,we have
![](https://www.scirp.org/html/5-7400449\7f5d5d77-1690-4d5d-be35-5f15483859e8.jpg)
Therefore all the zeros of F(z) whose modulus is greater than 1 lie in the circle.
![](https://www.scirp.org/html/5-7400449\1884ab46-b337-41d0-ba67-ac2e545f07cd.jpg)
But those zeros of F(z) whose modulus is less than or equal to 1 already satisfy the Inequality (9).
Since all the zeros of P(z) are also the zeros of F(z), therefore it follows that all the zeros of P(z) lie in the circle defined by (9). This completes the proof of Theorem 1.2.
PROOF OF THEOREM 1.3. Consider
![](https://www.scirp.org/html/5-7400449\3e52868b-a874-48f1-bc3d-8e6f284411e6.jpg)
therefore, for
we have
![](https://www.scirp.org/html/5-7400449\334c443e-e478-4e8e-a70b-e10b6350f4ad.jpg)
Proceeding similarly as in the proof of Theorem 1.2, we have
![](https://www.scirp.org/html/5-7400449\97490dec-d14f-4159-893c-68990deb086a.jpg)
therefore all the zeros of F(z) whose modules is greater than 1 lie in the circle
![](https://www.scirp.org/html/5-7400449\2ba2f9c7-eeff-479d-9538-83d84774100a.jpg)
But those zeros of F(z) whose modulus is
already satisfy the (11). Since all the zeros of P(z) are also the zero of F(z), therefore it follows that all the zeros of P(z) lie in circle defined by (11) and hence Theorem 1.3 is proved completed.
4. Acknowledgements
The authors are grateful to the refree for useful suggestions.