journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals, yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal

1. International Journal of Engineering Research and Development
e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com
Volume 8, Issue 8 (September 2013), PP. 53-61
www.ijerd.com 53 | Page
On The Zeros of a Polynomial in a Given Circle
M. H. Gulzar
Department of Mathematics University of Kashmir, Srinagar 190006
Abstract:- In this paper we discuss the problem of finding the number of zeros of a polynomial in a given circle
when the coefficients of the polynomial or their real or imaginary parts are restricted to certain conditions. Our
results in this direction generalize some well- known results in the theory of the distribution of zeros of
polynomials.
Mathematics Subject Classification: 30C10, 30C15
Keywords and phrases:- Coefficient, Polynomial, Zero.
I. INTRODUCTION AND STATEMENT OF RESULTS
In the literature many results have been proved on the number of zeros of a polynomial in a given circle.
In this direction Q. G. Mohammad [6] has proved the following result:
Theorem A: Let 



0
)(
j
j
j zazP be a polynomial o f degree n such that
0...... 011   aaaa nn .
Then the number of zeros of P(z) in
2
1
z does not exceed
0
log
2log
1
1
a
an
 .
K. K. Dewan [2] generalized Theorem A to polynomials with complex coefficients and proved the following
results:
Theorem B: Let 



0
)(
j
j
j zazP be a polynomial o f degree n such that jja )Re( , jja )Im( and
0...... 011    nn .
Then the number of zeros of P(z) in
2
1
z does not exceed
0
0
log
2log
1
1
a
n
j
jn 



.
Theorem C: Let 



0
)(
j
j
j zazP be a polynomial o f degree n with complex coefficients such that for some
real , ,
nja j ,......,2,1,0,
2
arg 


and
011 ...... aaaa nn   .
Then the number f zeros of P(z) in
2
1
z does not exceed

2. On The Zeros of a Polynomial in a Given Circle
www.ijerd.com 54 | Page
0
1
0
sin2)1sin(cos
log
2log
1
a
aa
n
j
jn 


 
.
The above results were further generalized by researchers in various ways.
M. H. Gulzar[4,5,6] proved the following results:
Theorem D: Let 



0
)(
j
j
j zazP be a polynomial o f degree n such that jja )Re( , jja )Im( and
0111 .......,......     knn ,
for .0,10,1 nk   Then the number of zeros of P(z) in 10,  z does not exceed
0
0
000 2)(2))(1(
log
1
log
1
a
k
n
j
jnn 
 


.
Theorem E: Let 



0
)(
j
j
j zazP be a polynomial o f degree n such that jja )Re( , jja )Im( and
011 ......   nn ,
for some ,10,0   ,then the number f zeros of P(z) in 10,  z , does not exceed
0
0
000 22)(2
log
1
log
1
a
n
j
jnn 
 

.
Theorem F: Let 



0
)(
j
j
j zazP be a polynomial o f degree n with complex coefficients such that for some
real , ,
nja j ,......,2,1,0,
2
arg 


and
011 ...... aaaa nn    ,
for some ,0 ,then the number of zeros of P(z) in 10,  z , does not exceed
0
1
1
0 )1sin(cossin2)1sin)(cos(
log
1
log
1
a
aaa
n
j
jn 


 

.
The aim of this paper is to find a bound for the number of zeros of P(z) in a circle of radius not
necessarily less than 1. In fact , we are going to prove the following results:
Theorem 1: Let 



0
)(
j
j
j zazP be a polynomial of degree n such that jja )Re( , jja )Im( and
0111 .......,......     knn ,

3. On The Zeros of a Polynomial in a Given Circle
www.ijerd.com 55 | Page
for .0,10,1 nk   Then the number of zeros of P(z) in )1,0(  cR
c
R
z does not exceed
0
0
000
1
]2)(2))(1([
log
log
1
a
kR
c
n
j
jnn
n


  
for 1R
and
0
1
00000 ]2)())(1([
log
log
1
a
kRa
c
n
j
jnn 
  
for 1R .
Remark 1: Taking R=1 and

1
c in Theorem 1, it reduces to Theorem D .
If the coefficients ja are real i.e. jj  ,0 , then we get the following result from Theorem 1:
Corollary 1: Let 



0
)(
j
j
j zazP be a polynomial o f degree n such that
0111 .......,...... aaakaaaa nn    ,
for .0,10,1 nk   Then the number of zeros of P(z) in )1,0(  cR
c
R
z does not exceed
0
000
1
)](2))(1([
log
log
1
a
aaaaakaaR
c
nn
n


for 1R
and
0
0000 )]())(1([
log
log
1
a
aaaaakaaRa
c
nn  
for 1R .
Applying Theorem 1 to the polynomial –iP(z) , we get the following result:
Theorem 2: Let 



0
)(
j
j
j zazP be a polynomial o f degree n such that jja )Re( , jja )Im( and
0111 .......,......     knn ,
for .0,10,1 nk   Then the number of zeros of P(z) in )1,0(  cR
c
R
z does not exceed
0
0
000
1
]2)(2))(1([
log
log
1
a
kR
c
n
j
jnn
n


  
for 1R
and
0
1
00000 ]2)())(1([
log
log
1
a
kRa
c
n
j
jnn 
  

4. On The Zeros of a Polynomial in a Given Circle
www.ijerd.com 56 | Page
for 1R .
Taking n in Theorem 1, we get the following result :
Corollary 2: Let 



0
)(
j
j
j zazP be a polynomial o f degree n such that jja )Re( , jja )Im( and
011 .......   nnk ,
for .0,10,1 nk   Then the number of zeros of P(z) in )1,0(  cR
c
R
z does not exceed
0
0
000
1
]2)(2)([
log
log
1
a
kR
c
n
j
jnn
n


 
for 1R
and
0
0
0000 ]2)()([
log
log
1
a
kRa
c
n
j
jnn 
 
for 1R .
Theorem 3: Let 



0
)(
j
j
j zazP be a polynomial o f degree n such that jja )Re( , jja )Im( and
011 ......   nn ,
for some ,10,  ,then the number of zeros of P(z) in )1,0(  cR
c
R
z , does not exceed
0
0
000
1
]22)([
log
log
1
a
R
c
n
j
jnn
n


 
for 1R
and
0
0
0000 ]2)([
log
log
1
a
Ra
c
n
j
jnn 
 
for 1R .
Remark 2: Taking R=1 and

1
c in Theorem 3, it reduces to Theorem E .
If the coefficients ja are real i.e. jj  ,0 , then we get the following result from Theorem 3:
Corollary 3: Let 



0
)(
j
j
j zazP be a polynomial o f degree n such that
011 ...... aaaa nn    ,
for some .10,  Then the number of zeros of P(z) in )1,0(  cR
c
R
z , does not exceed
0
000
1
]2)([
log
log
1
a
aaaaaR
c
nn
n


for 1R

5. On The Zeros of a Polynomial in a Given Circle
www.ijerd.com 57 | Page
and
0
0000 ])([
log
log
1
a
aaaaaRa
c
nn  
Applying Theorem 3 to the polynomial –iP(z) , we get the following result:
Theorem 4: Let 



0
)(
j
j
j zazP be a polynomial o f degree n such that jja )Re( , jja )Im( and
011 ......   nn ,
for some .10,0   Then the number of zeros of P(z) in )1,0(  cR
c
R
z , does not exceed
0
0
000
1
]22)([
log
log
1
a
R
c
n
j
jnn
n


 
for 1R
and
0
0
0000 ]2)([
log
log
1
a
Ra
c
n
j
jnn 
 
for 1R .
Taking 1,)1(  kk n in Corollary 3 , we get the following result:
Corollary 4: Let 



0
)(
j
j
j zazP be a polynomial o f degree n such that jja )Re( , jja )Im( and
011 ......   nnk ,
for some .10,  Then the number of zeros of P(z) in )1,0(  cR
c
R
z , does not exceed
0
0
000
1
]22)()([
log
log
1
a
kR
c
n
j
jnn
n


 
for 1R
and
0
0
0000 ]2)()([
log
log
1
a
kRa
c
n
j
jnn 
 
for 1R .
Theorem 5: Let 



0
)(
j
j
j zazP be a polynomial o f degree n such that
011 ...... aaaa nn   
for some .10,  Then the number of zeros of P(z) in )1,0(  cR
c
R
z , does not exceed
}]2)1sin(cos)1sin)(cos{([
1
log
log
1
00
1
0
 
n
n
aR
ac

6. On The Zeros of a Polynomial in a Given Circle
www.ijerd.com 58 | Page
for 1R
and
}])1sin(cos)1sin)(cos{([
1
log
log
1
000
0
aaaRa
ac
n  
for 1R .
For different values of the parameters ,,k in the above results, we get many other interesting results.
II. LEMMAS
For the proofs of the above results we need the following results:
Lemma 1: If f(z) is analytic in Rz  ,but not identically zero, f(0)  0 and
nkaf k ,......,2,1,0)(  , then
 

n
j j
i
a
R
fdf
1
2
0
log)0(log(Relog
2
1  


.
Lemma 1 is the famous Jensen’s theorem (see page 208 of [1]).
Lemma 2: If f(z) is analytic and )()( rMzf  in rz  , then the number of zeros of f(z) in 1,  c
c
r
z
does not exceed
)0(
)(
log
log
1
f
rM
c
.
Lemma 2 is a simple deduction from Lemma 1.
Lemma 3: Let 



0
)(
j
j
j zazP be a polynomial o f degree n with complex coefficients such that for some
real , , ,0,
2
arg nja j 

 and ,0,1 njaa jj   then any t>0,
 sin)(cos)( 111   jjjjjj aataatata .
Lemma 3 is due to Govil and Rahman [4].
III. PROOFS OF THEOREMS
Proof of Theorem 1: Consider the polynomial
F(z) =(1-z)P(z)
)......)(1( 01
1
1 azazazaz n
n
n
n  

0011
1
)(......)( azaazaaza n
nn
n
n  

1
110
1
)(......)( 


 
  zzaza n
nn
n
n






n
j
j
jj ziz
zzkk
0
10001
1
211
)()]()[(
......)(])1()[(

 



For Rz  , we have by using the hypothesis
0
1
)( aRazF n
n  
+



  RkRRn
nn 1
1
11 ...... 

 
RRRRk 001
1
21 )1(......)1(  


  



n
j
jj
1
1 )( 

7. On The Zeros of a Polynomial in a Given Circle
www.ijerd.com 59 | Page
  )1(......[ 1110
1
 

kkRaRa nn
nn
n
]2)1(......
1
1
000121 


 
n
j
jn  
nn
nn
n kRaRa    
00000
1
)())(1([
]2
1
1




n
j
j
000
1
2)())(1([    
kR nn
n
]2
0


n
j
j for 1R
and


n
j
jnn kRazF
1
00000 ]2)())(1([)(  
for 1R .
Therefore , by Lemma 3, it follows that the number of zeros of F(z) and hence
P(z) in )0,0(  cR
c
R
z does not exceed
0
0
000
1
2)(2))(1([
log
log
1
a
kR
c
n
j
jnn
n


  
for 1R
and
0
1
00000 2)())(1([
log
log
1
a
kRa
c
n
j
jnn 
  
for 1R .
That proves Theorem 1.
Proof of Theorem 3: Consider the polynomial
F(z) =(1-z)P(z)
)......)(1( 01
1
1 azazazaz n
n
n
n  

0011
1
)(......)( azaazaaza n
nn
n
n  

2
1210
1
)(......)( zzzaza n
nn
nn
n   



n
j
j
jj ziz
0
10001 )()]()[(  .
For Rz  , we have by using the hypothesis
0
1
)( aRazF n
n  
+ RRRR n
nn
n
01
2
121 ......   
+
j
n
j
jj RR 

0
10 )()1( 
011210
1
......[   

nnn
n
R


n
j
j
0
0 2)1(  ]

8. On The Zeros of a Polynomial in a Given Circle
www.ijerd.com 60 | Page
]22)([
0
000
1



n
j
jnn
n
R  for 1R
and


n
j
jnnRazF
0
0000 ]2)([)(  for 1R .
Therefore , by Lemma 2, it follows that the number of zeros of F(z) and hence
P(z) in )0,0(  cR
c
R
z does not exceed
0
0
000
1
]22)([
log
log
1
a
R
c
n
j
jnn
n


 
for 1R
and
0
0
0000 ]2)([
log
log
1
a
Ra
c
n
j
jnn 
 
for 1R .
That proves Theorem 3.
Proof of Theorem 5: Consider the polynomial
F(z) =(1-z)P(z)
)......)(1( 01
1
1 azazazaz n
n
n
n  

0011
1
)(......)( azaazaaza n
nn
n
n  

2
1210
1
)(......)( zaazaazaza n
nn
nn
n  


zaaaa )]()[( 0001   .
For Rz  , we have by using the hypothesis and Lemma 3
0
1
)( aRazF n
n  
+ RaaRaaRaaR n
nn
n
01
2
121 ......   
+ R0)1( 
n
nnnn
nn
n RaaaaRaRa ]sin)(cos)[( 110
1
 


Raaaa
RaRaaaa
]sin)(cos)[(
)1(]sin)(cos)[(......
0101
2
0
2
1212




]2)1sin(cos)1sin)(cos[( 00
1
aaR n
n
 

for 1R
and
])1sin(cos)1sin)(cos[( 000 aaaRa n  
for 1R .
Hence , by Lemma 2, it follows that the number of zeros of F(z) and therefore P(z)
in )0,0(  cR
c
R
z does not exceed
}]2)1sin(cos)1sin)(cos{([
1
log
log
1
00
1
0
 
n
n
aR
ac

9. On The Zeros of a Polynomial in a Given Circle
www.ijerd.com 61 | Page
for 1R
and
}])1sin(cos)1sin)(cos{([
1
log
log
1
000
0
aaaRa
ac
n  
for 1R .
That completes the proof of Theorem 5.
REFERENCES
[1]. L.V.Ahlfors, Complex Analysis, 3rd
edition, Mc-Grawhill
[2]. K. K. Dewan, Extremal Properties and Coefficient estimates for Polynomials with restricted Zeros and
on location of Zeros of Polynomials, Ph.D. Thesis, IIT Delhi, 1980.
[3]. N. K. Govil and Q. I. Rahman, On the Enestrom- Kakeya Theorem, Tohoku Math. J. 20(1968),126-
136.
[4]. M. H. Gulzar, On the Zeros of a Polynomial inside the Unit Disk, IOSR Journal of Engineering, Vol.3,
Issue 1(Jan.2013) IIV3II, 50-59.
[5]. M. H. Gulzar, On the Number of Zeros of a Polynomial in a Prescribed Region, Research Journal of
Pure Algebra-2(2), 2012, 35-49.
[6]. M. H. Gulzar, On the Zeros of a Polynomial, International Journal of Scientific and Research
Publications, Vol. 2, Issue 7,July 2012,1-7.
[7]. Q. G. Mohammad, On the Zeros of Polynomials, Amer. Math. Monthly 72(1965), 631-633.