http://www.ijeijournal.com/pages/v2i3.html

1. International Journal of Engineering Inventions
e-ISSN: 2278-7461, p-ISBN: 2319-6491
Volume 2, Issue 3 (February 2013) PP: 49-58
www.ijeijournal.com P a g e | 49
Further Generalizations of Enestrom-Kakeya Theorem
M. H. Gulzar
Department of Mathematics, University of Kashmir, Srinagar 190006
Abstract:- Many generalizations of the Enestrom –Kakeya Theorem are available in the literature. In this paper
we prove some results which further generalize some known results.
Mathematics Subject Classification: 30C10,30C15
Keywords and phrases:- Complex number, Polynomial , Zero, Enestrom – Kakeya Theorem.
I. INTRODUCTION AND STATEMENT OF RESULTS
The Enestrom –Kakeya Theorem (see[6]) is well known in the theory of the distribution of zeros of
polynomials and is often stated as follows:
Theorem A: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n whose coefficients satisfy
0...... 011   aaaa nn .
Then P(z) has all its zeros in the closed unit disk 1z .
In the literature there exist several generalizations and extensions of this result. Joyal et al [5] extended
it to polynomials with general monotonic coefficients and proved the following result:
Theorem B: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n whose coefficients satisfy
011 ...... aaaa nn   .
Then P(z) has all its zeros in
n
n
a
aaa
z
00 
 .
Aziz and zargar [1] generalized the result of Joyal et al [6] as follows:
Theorem C: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n such that for some 1k
011 ...... aaaka nn   .
Then P(z) has all its zeros in
n
n
a
aaka
kz
00
1

 .
For polynomials ,whose coefficients are not necessarily real, Govil and Rahman [2] proved the following
generalization of Theorem A:
Theorem C: If 

n
j
j
j zazP
0
)( is a polynomial of degree n with jja )Re( and jja )Im( ,
j=0,1,2,……,n, such that
0...... 011    nn ,
where 0n , then P(z) has all its zeros in
))(
2
(1
0


n
j
j
n
z 

.
Govil and Mc-tume [3] proved the following generalisations of Theorems B and C:

2. Further Generalizations of Enestrom-Kakeya Theorem
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Theorem D: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n with jja )Re( and jja )Im( ,
j=0,1,2,……,n. If for some 1k ,
011 ......   nnk ,
then P(z) has all its zeros in
n
n
j
jnk
kz

 


0
00 2
1 .
Theorem E: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n with jja )Re( and jja )Im( ,
j=0,1,2,……,n. If for some 1k ,
011 ......   nnk ,
then P(z) has all its zeros in
n
n
j
jnk
kz

 


0
00 2
1 .
M. H. Gulzar [4] proved the following generalizations of Theorems D and E:
Theorem F: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n with jja )Re( and jja )Im( ,
j=0,1,2,……,n. If for some real numbers 0 , ,10  
011 ......   nn ,
then P(z) has all its zeros in the disk
n
n
j
jn
n
z







0
000 2)(2
.
Theorem G: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n with jja )Re( and jja )Im( ,
j=0,1,2,……,n. If for some real number 0 ,
011 ......   nn ,
then P(z) has all its zeros in the disk
n
n
j
jn
n
z







0
000 2)(2
.
In this paper we give generalization of Theorems F and G. In fact, we prove the following:
Theorem 1: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n with jja )Re( and jja )Im( ,
j=0,1,2,……,n. If for some real numbers  , 0 , ,10,0,1   knank
01111 ......    knknknnn ,
and ,1 knkn    then P(z) has all its zeros in the disk
n
n
j
jknknn
n aa
z

 

0
000 2(21)1( 

,
and if ,1  knkn  then P(z) has all its zeros in the disk

3. Further Generalizations of Enestrom-Kakeya Theorem
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n
n
j
jknknn
n aa
z

 

0
000 2(21)1( 

Remark 1: Taking 1 , Theorem 1 reduces to Theorem F.
If ja are real i.e. 0j for all j, we get the following result:
Corollary 1: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n. If for some real numbers  , 0 ,
,0,1  knank ,10  
01111 ...... aaaaaaa knknknnn    ,
and ,1 knkn aa   then P(z) has all its zeros in the disk
n
knknn
n a
aaaaaa
a
z
)(21)1( 000 

 
,,
and if ,1  knkn aa then P(z) has all its zeros in the disk
n
knknn
n a
aaaaaa
a
z
)(21)1( 000 

 
If we apply Theorem 1 to the polynomial –iP(z) , we easily get the following result:
Theorem 2: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n with jja )Re( and jja )Im( ,
j=0,1, 2,……,n. If for some real numbers  , 0 , ,0,1  knank ,10  
01111 ......    knknknnn ,
and ,1 knkn    then P(z) has all its zeros in the disk
n
n
j
jknknn
n aa
z

 

0
000 2)(21)1( 

,
and if ,1  knkn  then P(z) has all its zeros in the disk
n
n
j
jknknn
n aa
z

 

0
000 2)(21)1( 

Remark 2: Taking 1 , Theorem 2 reduces to Theorem G.
Theorem 3: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n such that for some real numbers  , 0 ,
,10,,0,1   knank ,
01111 ...... aaaaaaa knknknnn   
and
.,......,1,0,
2
arg nja j 


If knkn aa  1 (i.e. 1 ) , then P(z) has all its zeros in the disk

4. Further Generalizations of Enestrom-Kakeya Theorem
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n
n
knjj
j
knn
n a
aaa
aa
a
z







1
,1
00 ]sin22)1cos(sin
)1sincossin(cos)sin(cos[



If 1  knkn aa (i.e. 1 ), then P(z) has all its zeros in the disk
n
n
knjj
j
knn
n a
aaa
aa
a
z







1
,1
00 ]sin22)1cos(sin
)1sincossin(cos)sin(cos[



Remark 3: Taking 1 in Theorem 3 , we get the following result:
Corollary 2: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n. If for some real numbers 0 , ,10  
011 ...... aaaa nn    ,
then P(z) has all its zeros in the disk
n
n
j
jn
n a
aaaa
a
z





1
,1
00 ]sin22)1cos(sin)sin(cos[ 

.
Remark 4: Taking 1,)1(  kak n in Cor.2, we get the following result:
Corollary 3: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n. If for some real numbers 0 , ,10  
011 ...... aaaak nn   ,
then P(z) has all its zeros in the disk
n
n
j
jn
n a
aaaak
a
z





1
,1
00 ]sin22)1cos(sin)sin(cos[ 

Taking 1,)1(  kak n , and 1 in Cor. 3, we get a result of Shah and Liman [7,Theorem 1].
II. LEMMAS
For the proofs of the above results , we need the following results:
Lemma 1: . Let P(z)= 
n
j
j
j za
0
be a polynomial of degree n with complex coefficients such that
fornjaj ,,...1,0,
2
arg 

 some real . Then for some t>0,
    .sincos 111  
  j
aataatata jjjjj
The proof of lemma 1 follows from a lemma due to Govil and Rahman [2].
Lemma 2.If p(z) is regular ,p(0)  0 and Mzp )( in ,1z then the number of zeros of p(z)in
,10,  z does not exceed
)0(
log
1
log
1
p
M

(see[8],p171).

5. Further Generalizations of Enestrom-Kakeya Theorem
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III. PROOFS OF THEOREMS
Proof of Theorem 1: Consider the polynomial
)()1()( zPzzF 
kn
knkn
kn
knkn
n
nn
n
n
n
n
n
n
zaazaazaaza
azazazaz









)()(......)(
)......)(1(
1
1
11
1
01
1
1
001
1
21 )(......)( azaazaa kn
knkn  

1
11
1
)(......)()( 


 kn
knkn
n
nn
n
nn zzzi 
0
1
100
01
1
211
)()1(
)(......)()(


iziz
zzz
n
j
j
jj
kn
knkn
kn
knkn








If ,1 knkn    then ,1 knkn    and we have
1
11
1
)(......)()()( 


 kn
knkn
n
nn
nn
nn zzzzizF 
......)()1()( 1
211  





kn
knkn
kn
kn
kn
knkn zzaz 
.)()1()( 0
1
10001  izizz
n
j
j
jj  

For 1z ,
1
1
1
211
1
)(......)()()( 




 kn
knkn
n
nn
n
nn
nn
n zzzzzazF 
0
1
10001
1
211
)()1()(
......)()1()(


izizz
zzaz
n
j
j
jj
kn
knkn
kn
kn
kn
knkn













  11211
1
)(......
1
)()( kknknnnnnn
n
zz
zaz 
n
n
j
jnjjnnn
knknkknkknkn
z
i
z
i
zzz
kzz
0
1
1
0
1
0
101
211
1
)()1(
1
)(
....
1
1
)(
1
)1(
1
)(












 knknnnnnn
n
zaz    1211 ......
 0
1
100
01211
1
......1







n
j
jj
knknknknkn
 11211 ......    knnknknnnnnn
n
kn
zaz 
 



n
j
j
knknkn
0
0
00121
2
)1(......1



6. Further Generalizations of Enestrom-Kakeya Theorem
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




n
j
j
knknnn
n
zaz
0
000
2
2)(1)1({[


>0
if

 
n
j
jknknnn za
0
000 22)(1)1( 
This shows that the zeros of F(z) whose modulus is greater than 1 lie in
n
n
j
jknknn
n aa
z

 

0
000 2)(21)1( 

.
But the zeros of F(z) whose modulus is less than or equal to 1 already satisfy the above inequality. Hence all the
zeros of F(z) lie in
n
n
j
jknknn
n aa
z

 

0
000 2)(21)1( 

.
Since the zeros of P(z) are also the zeros of F(z), it follows that all the zeros of P(z) lie in
n
n
j
jknknn
n aa
z

 

0
000 2)(21)1( 

.
If ,1  knkn  then ,1  knkn  and we have
1
11
1
)(......)()()( 


 kn
knkn
n
nn
nn
nn zzzzizF 
.)()1()(
......)()1()(
0
1
10001
1
21
1
1


izizz
zzz
n
j
j
jj
kn
knkn
kn
kn
kn
knkn










For 1z ,
1
1
1
211
1
)(......)()()( 




 kn
knkn
n
nn
n
nn
nn
n zzzzzazF 
0
1
10001
1
21
1
1
)()1()(
......)()1()(


izizz
zzz
n
j
j
jj
kn
knkn
kn
kn
kn
knkn













  11211
1
)(......
1
)()( kknknnnnnn
n
zz
zaz 
n
n
j
jnjjnnn
kknknkknkknkn
z
i
z
i
zzz
zzz
0
1
1
0
1
0
101
12111
1
)()1(
1
)(
....
1
)(
1
)1(
1
)(










 knknnnnnn
n
zaz    1211 ......

7. Further Generalizations of Enestrom-Kakeya Theorem
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 0
1
100
01211
1
......1







n
j
jj
knknknknkn
 11211 ......    knnknknnnnnn
n
kn
zaz 
 



n
j
j
knknkn
0
0
00121
2
)1(......1

















n
j
j
knknnn
n
za
z
0
0
00
}22
)(1)1({


>0
if

 
n
j
jknknnn za
0
000 22)(1)1( 
This shows that the zeros of F(z) whose modulus is greater than 1 lie in
n
n
j
jknknn
n aa
z

 

0
000 2)(21)1( 

.
But the zeros of F(z) whose modulus is less than or equal to 1 already satisfy the above inequality. Hence all the
zeros of F(z) lie in
n
n
j
jknknn
n aa
z

 

0
000 2)(21)1( 

.
Since the zeros of P(z) are also the zeros of F(z), it follows that all the zeros of P(z) lie in
n
n
j
jknknn
n aa
z

 

0
000 2)(21)1( 

.
That proves Theorem 1.
Proof of Theorem 3: Consider the polynomial
)()1()( zPzzF 
kn
knkn
kn
knkn
n
nn
n
n
n
n
n
n
zaazaazaaza
azazazaz









)()(......)(
)......)(1(
1
1
11
1
01
1
1
001
1
21 )(......)( azaazaa kn
knkn  
 .
If knkn aa  1 , then knkn aa  1 , 1 and we have, for 1z , by using Lemma1,
1
1
1
211
1
)(......)()()( 




 kn
knkn
n
nn
n
nn
nn
n zaazaazaazzazF 
0001
1
211
)1()(
......)()1()(
azazaa
zaazazaa kn
knkn
kn
kn
kn
knkn

 








8. Further Generalizations of Enestrom-Kakeya Theorem
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


  11211
1
)(......
1
)()( kknknnnnnn
n
z
aa
z
aaaazaz 
nnn
kknknkknkknkn
z
a
z
a
z
aa
z
aa
z
a
z
aa
0
1
0
101
1211
)1(
1
)(
....
1
)(
1
)1(
1
)(






 knknnnnnn
n
aaaaaazaz   1211 ......
 00
01211
1
......1
aa
aaaaaaa knknknknkn

 


  sin)(cos){( 11   nnnnn
n
aaaazaz
 cos)(......sin)(cos)( 12121 knknnnnn aaaaaa  
(  sin)(cos)(sin) 111   knknknknknkn aaaaaa
 sin)(cos)(1 2121   knknknknkn aaaaa
})1(sin)(cos)(...... 000101 aaaaaa  
  sincossin(cos)sin(cos{  knnn
n
aazaz
}sin22)1cos(sin)1
1
,1
00 



n
knjj
jaaa 
0
if
  sincossin(cos)sin(cos  knnn aaza




1
,1
00 sin22)1cos(sin)1
n
knjj
jaaa 
This shows that the zeros of F(z) whose modulus is greater than 1 lie in
n
n
knjj
j
knn
n a
aaa
aa
a
z







1
,1
00 ]sin22)1cos(sin
)1sincossin(cos)sin(cos[



. .
But the zeros of F(z) whose modulus is less than or equal to 1 already satisfy the above inequality. Hence all the
zeros of F(z) lie in
n
n
knjj
j
knn
n a
aaa
aa
a
z







1
,1
00 ]sin22)1cos(sin
)1sincossin(cos)sin(cos[



.
Since the zeros of P(z) are also the zeros of F(z), it follows that all the zeros of P(z) lie in
n
n
knjj
j
knn
n a
aaa
aa
a
z







1
,1
00 ]sin22)1cos(sin
)1sincossin(cos)sin(cos[




9. Further Generalizations of Enestrom-Kakeya Theorem
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If 1  knkn aa , then 1  knkn aa , 1 and we have, for 1z , by using Lemma 1,
1
1
1
211
1
)(......)()()( 




 kn
knkn
n
nn
n
nn
nn
n zaazaazaazzazF 
0001
1
21
1
1
)1()(
......)()1()(
azazaa
zaazazaa kn
knkn
kn
kn
kn
knkn

 










  11211
1
)(......
1
)()( kknknnnnnn
n
z
aa
z
aaaazaz 
nnn
kknknkknkknkn
z
a
z
a
z
aa
z
aa
z
a
z
aa
0
1
0
101
12111
)1(
1
)(
....
1
)(
1
)1(
1
)(






 knknnnnnn
n
aaaaaazaz    1211 ......
 00
01211
1
......1
aa
aaaaaaa knknknknkn

 


  sin)(cos){( 11   nnnnn
n
aaaazaz
 cos)(......sin)(cos)( 12121 knknnnnn aaaaaa  
(  sin)(cos)(sin) 111   knknknknknkn aaaaaa
 sin)(cos)(1 2121   knknknknkn aaaaa
})1(sin)(cos)(...... 000101 aaaaaa  
  sincossin(cos)sin(cos{  knnn
n
aazaz
}sin22)1cos(sin)1
1
,1
00 



n
knjj
jaaa 
0
if
  sincossin(cos)sin(cos  knnn aaza




1
,1
00 sin22)1cos(sin)1
n
knjj
jaaa 
This shows that the zeros of F(z) whose modulus is greater than 1 lie in
n
n
knjj
j
knn
n a
aaa
aa
a
z







1
,1
00 ]sin22)1cos(sin
)1sincossin(cos)sin(cos[



.
But the zeros of F(z) whose modulus is less than or equal to 1 already satisfy the above inequality. Hence all the
zeros of F(z) and therefore P(z) lie in
n
n
knjj
j
knn
n a
aaa
aa
a
z







1
,1
00 ]sin22)1cos(sin
)1sincossin(cos)sin(cos[



That proves Theorem 3.

10. Further Generalizations of Enestrom-Kakeya Theorem
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Vol.11,No.3,2002, 246-253.
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5) A. Joyal, G. Labelle, Q.I. Rahman, On the location of zeros of polynomials, Canadian Math.
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