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1. International Journal of Engineering Research and Development
e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com
Volume 6, Issue 11 (April 2013), PP. 68-75
68
On The Zero-Free Regions for Polynomials
M. H. GULZAR
Department of Mathematics University of Kashmir, Srinagar 190006
Abstract: In this paper we find the zero-free regions for a class of polynomials whose coefficients or
their real and imaginary parts are restricted to certain conditions.
Mathematics Subject Classification: 30C10, 30C15
Keywords and phrases:- Coefficient, Polynomial, Zero.
I. INTRODUCTION AND STATEMENT OF RESULTS
The following result, known as the Enestrom-Kakeya Theorem [5] , is well-known in the theory of distribution
of zeros of polynomials:
Theorem A: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n such that
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. Recently, Choo and Choi [1]
proved the following results:
Theorem B: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n such that
either 132 ...... aaaa nn   and 0231 ...... aaaa nn   ,if n is odd , or
022 ...... aaaa nn   and 1331 ...... aaaa nn   , if n is even. Then P(z)
does not vanish in
01111
0
aaaaaaa
a
z
nnnn 


.
Theorem C: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n with jja )Re( and jja )Im( ,
j=0,1,2,……,n, such that
011 ......   nnk ,
011 ......   nn .
Then P(z) does not vanish in
00
0
)1(  

nnnn kka
a
z .
Theorem D: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n such that for some  ,
2
arg

 ja , j=0,1,……,n,
and for some 1k ,
011 ...... aaaak nn   .
Then P(z) does not vanish in

2. On The Zero-Free Regions for Polynomials
69




 1
1
00
0
sin2sin)(cos)(
n
j
jnnn aaakaakak
a
z

.
M. H. Gulzar [3,4] proved the following results:
Theorem E: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n such that for some 11 k , 12 k ,
10 1  , 10 2  ,either 11321 ...... aaaak nn   and 022312 ...... aaaak nn   ,if
n is odd , or 01221 ...... aaaak nn   and 123312 ...... aaaak nn   , if n is even. Then
for odd n, P(z) has all its zeros in
nn
n
a
K
a
a
z 11
 
,
where
)()()(2)()( 1111011121`1 aaaaaaaakaakK nnnnnn   
)( 002 aa  ,
and for even n, P(z) has all its zeros in
nn
n
a
K
a
a
z 21
 
,
where
)()()(2)()( 0011011121`2 aaaaaaaakaakK nnnnnn   
)( 112 aa  .
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, such that
011 ......   nnk .
Then all the zeros of P(z) lie in
n
n
j
jnn
n
n
a
k
a
kz





1
0
000 2)(2
)1(


.
Theorem G: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n such that for some  ,
2
arg

 ja , j=0,1,……,n,
and for some 1k , 10 
011 ...... aaaak nn   .
Then P(z) has all its zeros in






 


1
1
000 sin2)(2)sin(cos
1
1
n
j
jn
n
aaaaak
a
kz  .
The aim of this paper is to generalize some of the above-mentioned results. More precisely, we prove the
following results:

3. On The Zero-Free Regions for Polynomials
70
Theorem 1: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n such that for some 11 k , 12 k ,
10 1  , 10 2  ,either 11321 ...... aaaak nn   and 022312 ...... aaaak nn   ,if
n is odd , or 01221 ...... aaaak nn   and 123312 ...... aaaak nn   , if n is even. Then
P(z) does not vanish in
010021111121
0
2)()()()( aaaaaaaakaak
a
z
nnnn 

 
, if n is odd,
and in
011120011121
0
2)()()()( aaaaaaaakaak
a
z
nnnn 

 
,if n is even.
Remark 1: Taking 11 k , 12 k , 11  , 12  , Theorem 1 reduces to Theorem B.
If z is a zero of 

n
j
j
j zazP
0
)( , then
n
n
n
n
n
n
n
n
a
a
a
a
z
a
a
a
a
zz 1111 
 .
Therefore, combining Theorem E and Theorem 1, we arrive at the following result:
Corollary 1: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n such that for some 11 k , 12 k ,
10 1  , 10 2  ,either 11321 ...... aaaak nn   and 022312 ...... aaaak nn   ,if
n is odd , or 01221 ...... aaaak nn   and 123312 ...... aaaak nn   , if n is even. Then
for odd n, P(z) has all its zeros in
010021111121
0
2)()()()( aaaaaaaakaak
a
nnnn   
z
 )()()(2)()(
1
1111011121` aaaaaaaakaak
a
nnnnnn
n
  
1002 )(  naaa ,
and for even n, P(z) has all its zeros in
011120011121
0
2)()()()( aaaaaaaakaak
a
nnnn   
z
 )()()(2)()(
1
0011011121` aaaaaaaakaak
a
nnnnnn
n
  
1112 )(  naaa .
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, such that
011 ......   nnk .
Then P(z) does not vanish in

4. On The Zero-Free Regions for Polynomials
71




 1
1
0000
0
2)()(
n
j
jnnnn ka
a
z

.
Remark 2: If in Theorem 2, we have, in addition,
011 ......   nn ,
then 0
1
 
 n
n
j
jj , and we have the following result:
Theorem 3: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n with jja )Re( and jja )Im( ,
j=0,1,2,……,n, such that
011 ......   nnk ,
011 ......   nn .
Then P(z) does not vanish in
0000
0
)()(  

nnnn ka
a
z .
Remark 3: Taking 1 , Theorem 3 reduces to Theorem C.
Combining Theorem 2 and Theorem F, we get the following result:
Corollary 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, such that
011 ......   nnk .
Then P(z) has all its zeros in




1
1
0000
0
2)()(
n
j
jnnnn ka
a

z
])1(2)(2[
1
1
1
0
000 


  n
n
j
jnn
n
akk
a

Theorem 4: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n such that for some  ,
2
arg

 ja , j=0,1,……,n,
and for some 1k , 10 
011 ...... aaaak nn   .
Then P(z) does not vanish in




 1
1
00
0
sin2)1sin(cos)1sin(cos
n
j
jn aaaak
a
z

.
Remark 4: Taking 1 , Theorem 4 reduces to Theorem D.
Combining Theorem 4 and Theorem G, we get the following result:

5. On The Zero-Free Regions for Polynomials
72
Corollary 3 : Let 

n
j
j
j zazP
0
)( be a polynomial of degree n such that for some  ,
2
arg

 ja , j=0,1,……,n,
and for some 1k , 10 
011 ...... aaaak nn   .
Then P(z) has all its zeros in




1
1
00
0
sin2)1sin(cos)1sin(cos
n
j
jn aaaak
a

z






 


n
n
j
jn
n
akaaaaak
a
)1(sin2)(2)sin(cos
1 1
1
000  .
II. LEMMA
For the proofs of the above results, we need the following lemma:
Lemma: For any two complex numbers 0b and 1b such that 10 bb  and
1,0,
2
arg  jbj

 for some real  ,
 sin)(cos)( 101010 bbbbbb  .
The above lemma is due to Govil and Rahman [2].
III. PROOFS OF THE THEOREMS
Proof of Theorem 1: Let n be odd. Consider the polynomial
)()1()( 2
zPzzF 
......)()(
)......)(1(
1
312
1
1
2
01
1
1
2









n
nn
n
nn
n
n
n
n
n
n
n
n
zaazaazaza
azazazaz
01
2
02
3
13 )()( azazaazaa 
)(0 zqa  ,
where
......)()()( 1
312
1
1
2
 



 n
nn
n
nn
n
n
n
n zaazaazazazq
zazaazaa 1
2
02
3
13 )()( 
1
312121
1
1
2
)()1()( 




 n
nn
n
n
n
nn
n
n
n
n zaakzakzaakzaza
2
022
3
11
3
113
1
12 )()1()(......)1( zaazazaazak n
n   

zaza 1
2
02 )1(  
For 1z ,
......)1()1()( 123121111   nnnnnnnn akaakakaakaazq
10202211113 )1()1( aaaaaaa  
010021111121 2)()()()( aaaaaaaakaak nnnn   
1M .

6. On The Zero-Free Regions for Polynomials
73
Since q(z) is analytic for 1z and q(0)=0, it follows , by Rouche’s theorem, that
zMzq 1)(  for 1z .
Hence, it follows that, for 1z ,
)()( 0 zqazF 
zMa
zqa
10
0 )(


0
if
1
0
M
a
z  .
It is easy to see that 01 aM  and the proof is complete if n is odd.
The proof for even n is similar.
Proof of Theorem 2: Consider the polynomial
)()1()( zPzzF 
......)()(
)......)(1(
1
211
1
01
1
1







n
nn
n
nn
n
n
n
n
n
n
zaazaaza
azazazaz
001
2
12 )()( azaazaa 
2
12
1
211
1
).....()()( zzzza n
nn
n
nn
n
n   




n
j
j
jj ziaz
1
1001 )()( 
)(0 zqa  ,
where
.)(.....)()()( 2
12
1
211
1
zzzzazq n
nn
n
nn
n
n   




n
j
j
jj ziz
1
101 )()( 
1
211
1
)()1()( 


 n
nn
n
n
n
nn
n
n zzkzkza 


n
j
j
jj zizzz
1
1001
2
12 )()1()()(...... 
For 1z ,
12211 ......)1()(    nnnnnn kkazq
)()1(
1
1001 

n
j
jj 




1
1
0000 2)()(
n
j
jnnnnn ka 
2M
Since q(z) is analytic for 1z and q(0)=0, it follows , by Rouche’s theorem, that
zMzq 2)(  for 1z .
Hence, it follows that, for 1z ,

7. On The Zero-Free Regions for Polynomials
74
)()( 0 zqazF 
zMa
zqa
20
0 )(


0
if
2
0
M
a
z  .
It is easy to see that 02 aM  and the proof is complete
Proof of Theorem 4: Consider the polynomial
)()1()( zPzzF 
......)()(
)......)(1(
1
211
1
01
1
1







n
nn
n
nn
n
n
n
n
n
n
zaazaaza
azazazaz
001
2
12 )()( azaazaa 
)(0 zqa  ,
where
2
12
1
211
1
)(.....)()()( zaazaazaazazq n
nn
n
nn
n
n  


zaa )( 01 
1
211
1
)()1()( 


 n
nn
n
n
n
nn
n
n zaazakzakaza
zazaazaa 001
2
12 )1()()(......   .
For 1z , we have , by using the lemma,
12211 ......)1()( aaaaakakaazq nnnnnn  
001 )1( aaa  
nnnnnn akaakaaka )1(sin)(cos)( 11   
001
011212
2121
)1(sin)(
cos)(sin)(cos)(
......sin)(cos)(
aaa
aaaaaa
aaaa nnnn





 




1
1
00 sin2)1sin(cos)1sin(cos
n
j
jn aaaak 
3M
Since q(z) is analytic for 1z and q(0)=0, it follows , by Rouche’s theorem, that
zMzq 3)(  for 1z .
Hence, it follows that, for 1z ,
)()( 0 zqazF 
zMa
zqa
30
0 )(


0
if

8. On The Zero-Free Regions for Polynomials
75
3
0
M
a
z  .
It is easy to see that 03 aM  and the proof is complete
REFERENCES
[1]. Y. Choo and G. K. Choi, On the Zero-free Regions of Polynomials, Int.Journal of Math. Analysis,
Vol.5,2011,no. 20, 975-981.
[2]. N. K. Govil and Q. I. Rahman, On the Enestrom-Kakeya Theorem,Tohoku Math.J. 20(1968), 126-136.
[3]. M. H. Gulzar, Distribution of the Zeros of a Polynomial, Int. Journal Of Innovative Research and
Development, Vol.2 Issue 3, March 2013,416-427.
[4]. M. H. Gulzar Some Refinements of Enestrom-Kakeya Theorem, , Int.Journal of Mathematical Archive
-2(9), September 2011, 1512-1519.
[5]. M. Marden, Geometry of Polynomials, Math. Surveys No. 3, Amer.Math. Soc. (RI Providence) , 1966.