International Journal of Engineering Research and Development (IJERD)

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
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

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
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
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
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 ,

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
0111 .......,...... aaakaaaa nn    ,
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
0111 .......,......     knn ,
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 .
Taking n in Theorem 1, we get the following result :



0
)(
j
j
011 .......   nnk ,
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
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:



0
)(
j
j
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

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
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:



0
)(
j
j
011 ......   nnk ,
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
011 ...... aaaa nn   
c
R
z , does not exceed
}]2)1sin(cos)1sin)(cos{([
1
log
log
1
00
1
0
 
n
n
aR
ac

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
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 )( 

  )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.
F(z) =(1-z)P(z)
)......)(1( 01
1
1 azazazaz n
n
n
n  

0011
1
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(  ]

]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.
F(z) =(1-z)P(z)
)......)(1( 01
1
1 azazazaz n
n
n
n  

0011
1
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

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.

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