The document presents several theorems that generalize the Enestrom-Kakeya theorem about zero-free regions for polynomials with restricted coefficients. Theorem 1 shows that if the coefficients of a polynomial satisfy certain inequalities involving parameters k and ρ, then the polynomial does not vanish in a specified disk. Theorem 2 gives a similar result involving parameters ρ and τ. Theorem 3 is a special case of Theorem 2. The proofs involve manipulating inequalities involving the coefficients and applying the properties of zeros of polynomials.

1. Research Inventy: International Journal Of Engineering And Science
Issn: 2278-4721, Vol. 2, Issue 6 (March 2013), Pp 06-10
Www.Researchinventy.Com
6
Zero-Free Regions for Polynomials With Restricted Coefficients
M. H. Gulzar
Department of Mathematics University of Kashmir, Srinagar 190006
Abstract : According to a famous result of Enestrom and Kakeya, if
01
1
1 ......)( azazazazP n
n
n
n  

is a polynomial of degree n such that
011 ......0 aaaa nn   ,
then P(z) does not vanish in 1z . In this paper we relax the hypothesis of this result in several ways and
obtain zero-free regions for polynomials with restricted coefficients and thereby present some interesting
generalizations and extensions of the Enestrom-Kakeya Theorem.
Mathematics Subject Classification: 30C10,30C15
Key-Words and Phrases: Zero-free regions, Bounds, Polynomials
1. Introduction And Statement Of Results
The following elegant result on the distribution of zeros of a polynomial is due to Enestrom and
Kakeya [6] :
Theorem A : If 

n
j
j
j zazP
0
)( is a polynomial of degree n such that
0...... 011   aaaa nn ,
then P(z) has all its zeros in 1z .
Applying the above result to the polynomial )
1
(
z
Pzn
, we get the following result :
Theorem B : If 

n
j
j
j zazP
0
)( is a polynomial of degree n such that
011 ......0 aaaa nn   ,
then P(z) does not vanish in 1z .
In the literature [1-5, 7,8], there exist several extensions and generalizations of the Enestrom-Kakeya Theorem .
Recently B. A. Zargar [9] proved the following results:
Theorem C: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n . If for some real number 1k ,
011 ......0 kaaaa nn   ,
then P(z) does not vanish in the disk
12
1


k
z .
Theorem D: Let


n
j
j
j zazP
0
)( be a polynomial of degree n . If for some real number na  0, ,
011 ......0 aaaa nn   ,
then P(z) does not vanish in the disk

2. Zero-free Regions for Polynomials with Restricted Coefficients
7
0
2
1
1
a
z


 .
Theorem E: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n . If for some real number 1k ,
0...... 011   aaaka nn ,
then P(z) does not vanish in
0
0
2 aka
a
z
n 
 .
Theorem F: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n . If for some real number ,0 ,
0...... 011   aaaa nn  ,
then P(z) does not vanish in the disk
0
0
)(2 aa
a
z
n 


.
In this paper we give generalizations of the above mentioned results. In fact, we prove the following results:
Theorem 1: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n . If for some real numbers 1k and
,0 ,
011 ...... kaaaa nn   ,
then P(z) does not vanish in the disk
nn aaaaak
a
z


2)( 000
0
.
Remark 1: Taking ,0 na  Theorem 1 reduces to Theorem C and taking k=1 and ,0 na  , it
reduces to Theorem D.
Theorem 2: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n . If for some real numbers 0 and
10  ,
011 ...... aaaa nn    ,
then P(z) does not vanish in
000
0
)(2 aaaaa
a
z
nn 


.
Remark 2: Taking 1 and 00 a , Theorem 1 reduces to Theorem F and taking 1 , 00 a and
1,)1(  kak n , it reduces to Theorem E .
Also taking 1,)1(  kak n , we get the following result which reduces to Theorem E by taking 00 a
and 1 .
Theorem 3: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n . If for some real numbers 10,1  k ,
011 ...... aaaka nn   ,
then P(z) does not vanish in the disk

3. Zero-free Regions for Polynomials with Restricted Coefficients
8
0
0
)21(2 aka
a
z
n 
 .
2.Proofs of the Theorems
Proof of Theorem 1: We have
01
1
1 ......)( azazazazP n
n
n
n  
 .
Let
)
1
()(
z
PzzQ n

and
)()1()( zQzzF  .
Then
)......)(1()( 1
1
10 nn
nn
azazazazzF  

2
12
1
2110
1
0 )(......)()[( zaazaazaaza nn
nnn



])( 1 nnn azaa   .
For 1z ,
]......[)( 1
1
2110
1
0 nnn
nnn
azaazaazaazazF  

}]......{
1
[ 1
121
10
0
0 n
n
n
nnn
z
a
z
aa
z
aa
aa
a
zza 



 

   nn
n
aaaaakaaka
a
zza 1210010
0
0 ......{
1
[
}]na
)(......)()1(){(
1
[ 1221010
0
0   nn
n
aaaaakaka
a
zza
}])( 1 nnn aaa   
}]2)({
1
[ 000
0
0  nn
n
aaaaak
a
zza
0
if
]2)([
1
000
0
 nn aaaaak
a
z .
This shows that all the zeros of F(z) whose modulus is greater than 1 lie in the closed disk
]2)([
1
000
0
 nn aaaaak
a
z .
But those zeros of F(z) whose modulus is less than or equal to 1 already lie in the above disk. Therefore, it
follows that all the zeros of F(z) and hence Q(z) lie in
]2)([
1
000
0
 nn aaaaak
a
z .
Since )
1
()(
z
QzzP n
 , it follows, by replacing z by
z
1
, that all the zeros of P(z) lie in

4. Zero-free Regions for Polynomials with Restricted Coefficients
9
2)( 000
0


nn aaaaak
a
z .
Hence P(z) does not vanish in the disk
2)( 000
0


nn aaaaak
a
z .
That proves Theorem 1.
Proof of Theorem 2: We have
01
1
1 ......)( azazazazP n
n
n
n  
 .
Let
)
1
()(
z
PzzQ n

and
)()1()( zQzzF  .
Then
)......)(1()( 1
1
10 nn
nn
azazazazzF  

2
12
1
2110
1
0 )(......)()[( zaazaazaaza nn
nnn



])( 1 nnn azaa   .
For 1z ,
]......[)( 1
1
2110
1
0 nnn
nnn
azaazaazaazazF  

}]......{
1
[ 1
121
10
0
0 n
n
n
nnn
z
a
z
aa
z
aa
aa
a
zza 



 

   nn
n
aaaaaaaa
a
zza 1210010
0
0 ......{
1
[
}]na
)(......)()1(){(
1
[ 112001
0
0  nn
n
aaaaaaa
a
zza 
}]na 
}]2)({
1
[ 000
0
0   nn
n
aaaaa
a
zza
0
if
}2)({
1
000
0
  nn aaaaa
a
z .
This shows that all the zeros of F(z) whose modulus is greater than 1 lie in the closed disk
}2)({
1
000
0
  nn aaaaa
a
z .
But those zeros of F(z) whose modulus is less than or equal to 1 already lie in the above disk. Therefore, it
follows that all the zeros of F(z) and hence Q(z) lie in
}2)({
1
000
0
  nn aaaaa
a
z .

5. Zero-free Regions for Polynomials with Restricted Coefficients
10
Since )
1
()(
z
QzzP n
 , it follows, by replacing z by
z
1
, that all the zeros of P(z) lie in
 2)( 000
0


nn aaaaa
a
z .
Hence P(z) does not vanish in the disk
 2)( 000
0


nn aaaaa
a
z .
That proves Theorem 2.
References
[1] Aziz, A. and Mohammad, Q. G., Zero-free regions for polynomials and some generalizations of Enesrtrom-Kakeya Theorem,
Canad. Math. Bull., 27(1984), 265-272.
[2] Aziz, A. and Zargar B.A., Some Extensions of Enesrtrom-Kakeya Theorem, Glasnik Mathematicki, 31(1996) , 239-244.
[3] Dewan, K. K., and Bidkham, M., On the Enesrtrom-Kakeya Theorem I., J. Math. Anal. Appl., 180 (1993) , 29-36.
[4] Kurl Dilcher, A generalization of Enesrtrom-Kakeya Theorem, J. Math. Anal. Appl., 116 (1986) , 473-488.
[5] Joyal, A., Labelle, G..and Rahman, Q. I., On the Location of zeros of polynomials, Canad. Math. Bull., 10 (1967), 53-63.
[6] Marden, M., Geometry of Polynomials, IInd. Edition, Surveys 3, Amer. Math. Soc., Providence, (1966) R.I.
[7] Milovanoic, G. V., Mitrinovic , D.S., Rassiass Th. M., Topics in Polynomials, Extremal problems, Inequalities , Zeros, World
Scientific , Singapore, 1994.
[8] Rahman, Q. I., and Schmeisser, G., Analytic Theory of Polynomials, 2002, Clarendon Press, Oxford.
[9] Zargar, B. A., Zero-free regions for polynomials with restricted coefficients, International Journal of Mathematical Sciences and
Engineering Applications, Vol. 6 No. IV(July 2012), 33-42.