Keywords: complex polynomials, zeros, Eneström – Kakeya Theorem
American Journal of Mathematical Analysis, 2014 2 (2),
pp 2327.
DOI: 10.12691/ajma222
Received April 30, 2014; Revised May 19, 2014; Accepted May 25, 2014
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction and Statement of Results
The following result known as EneströmKakeya theorem, is well known in the theory of distribution of zeros of polynomials was firstly proved by Eneström ^{[2]} and Kakeya ^{[5]} and later independently by Hurwitz ^{[3]}.
Theorem 1.1. If is a polynomial of degree n such that
then all the zeros of lie in .
Joyal, Labelle and Rahman ^{[4]} extended Theorem A to the polynomial whose coefficients are monotonic but not necessarily nonnegative by proving the following:
Theorem 1.2. Let be a polynomial of degree n such that
then all the zeros of lie in
Aziz and Zargar ^{[6]} relaxed the hypothesis of EneströmKakeya theorem and proved the following extension of Theorem 1.2.
Theorem 1.3. Letbe a polynomial of degree n such that for some,
then all the zeros of lie in
By using Schwartz lemma, Aziz and Mohammad ^{[1]} generalized Eneström Kakeya theorem in a different way and proved:
Theorem 1.4. If be a polynomial of degree n with real positive coefficients. If can be found such that
then all the zeros of lie in .
In this paper, we prove some generalizations and extensions of Theorem1.1, Theorem 1.2, Theorem 1.3 and Theorem 1.4. In this direction we first present the following interesting result which is generalization of Theorem 1.4.
Theorem 1.5. If be a polynomial of degree n such that where and , j = 0, 1, ….., n are real numbers and for certain nonnegative real numbers with and
then all the zeros of lie in
where
Remark 1.1. If in Theorem 1.5, we assume that all the coefficients are real and positive, then M=0 and it reduces to Theorem 1.4 due to Aziz and Mohammad ^{[1]}.
In particular, if we choose t_{2}=0 and in Theorem 1.5, then we have the following result.
Corollary 1.1. If be a polynomial of degree n such that where and , j = 0, 1, 2 ….., n are real numbers. If be such that
then all the zeros of lie in
where
Remark 1.2. If we consider all the coefficients in Corollary 1.1 to be real and choose t = 1, then we get Theorem 1.2due to Joyal, Labelle and Rahman ^{[4]}. In addition, if we choose all coefficients to be positive, then we obtain the wellknown EneströmKakeya Theorem (Theorem 1.1).
Next we prove the following result which is also a generalization of Theorem
Theorem.1.6. Let be a polynomial of degree n such that where and , j = 0, 1, ….., n are real numbers and for certain nonnegative real numbers with and
and for some
and
then all the zeros of lie in
where
Remark 1.3. If we choose and in Theorem 1.6, we get the following result.
Corollary 1.2. Let be a polynomial of degree n such that where and , j = 0, 1, 2,….., n are real numbers and for certain nonnegative real number t
and for some ,
Then all the zeros of lie in
where
Remark 1.4. If we consider all the coefficients in Corollary 1.2 to be real and positive, then we obtain Theorem 1.3 due to Aziz and Zargar ^{[1]}.
Remark 1.5. Again, if we choose all the coefficients in Corollary 1 to be real and consider k=1 and t=1, we get Theorem B due to Joyal, Labelle and Rahman ^{[5]}.
2. Proof of the Theorem
Proof of Theorem1.5. Consider the polynomial
Further, let
 (3.1) 
where
Now
This gives after using hypothesis, for z = 1/t_{1}
Clearly
Therefore, by Schwarz’s Lemma
Hence from (3.1), if , then we get
If .
That is
Thus in .
Consequently, all the zeros of G(z) lie in.
As we conclude that all the zeros of F(z) and hence all the zeros of p(z) lie in
This completes the proof of the theorem 1.5.
Proof of Theorem 1.6. Consider the polynomial
This gives
Therefore for , we have
if
Hence all the zeros of whose modulus is greater than lie in the circle
Since all the zeros whose modulus is less than already lies in this circle, we conclude that all the zeros of and hence lie in
This completes the proof of the Theorem1.6.
References
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 In article  

[3]  A. Hurwitz, Űberdie Nullslenllen der Bessel Schen Function, Math. Ann., 33(1989)246266. 
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[6]  A. Aziz and B. A. Zargar, Some extensions of EnströmKakeya Theorem, Glasnik Mate., 31(1996), 239244. 
 In article  
