ABSTRACT: In this paper we find the number of zeros of a polynomial inside the unit disc under certain conditions on the coefficients of the polynomial. Mathematics Subject Classification: 30C10, 30C15.

1. International Journal of Engineering Research and Development
e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com
Volume 13, Issue 2 (February 2017), PP.67-70
67
On the Zeros of A Polynomial Inside the Unit Disc
M.H. Gulzar
Department Of Mathematics, University Of Kashmir, Srinagar
ABSTRACT: In this paper we find the number of zeros of a polynomial inside the unit disc under certain
conditions on the coefficients of the polynomial.
Mathematics Subject Classification: 30C10, 30C15.
Keywords and Phrases: Coefficients, Polynomial, Zeros.
I. INTRODUCTION
In the context of the Enestrom-Kakeya Theorem [4] which states that all the zeros of a polynomial


n
j
j
j zazP
0
)( with 0...... 011   aaaa nn lie in 1z , Q. G. Mohammad [ 5] proved the
following result giving a bound for the number of zeros of P(z) in
2
1
z :
Theorem A: Let 

n
j
j
j zazP
0
)( be a polynomial of 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
 .
Various bounds for the number of zeros of a polynomial with certain conditions on the coefficients were
afterwards given by researchers in the field (e.g. see [1],[2],[3]).
II. MAIN RESULTS
In this paper we find a bound for the number of zeros of a polynomial in a closed disc of radius less
than 1 and prove
Theorem 1: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n with ,)Im(,)Re( jjjj aa  
nj ,......,2,1,0 such that for some 10,  n and for some 1,1  ok ,
    11 ......nnk
and
001211 ......    L .
Then the number of zeros of P(z) in 10,  z does not exceed
0
0
2)1()1(
log
1
log
1
a
Lkka
n
j
jnnn 
 


.
Taking ja real i.e. njj ,.....,2,1,0,0  , Theorem 1 reduces to the following result:
Corollary 1: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n such that for some 10,  n and for
some 1,1  ok ,
 aaaka nn   11 ......

2. On The Zeros Of A Polynomial Inside The Unit Disc
68
and
001211 ...... aaaaaaaL   
Then the number f zeros of P(z) in 10,  z does not exceed
0
)1()(
log
1
log
1
a
aLaaak nn  


.
Taking 1 in Cor. 1 , we get the following result:
Corollary 2: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n such that for some 10,  n and for
some ,1k
 aaaka nn   11 ......
and
001211 ...... aaaaaaaL   
Then the number f zeros of P(z) in 10,  z does not exceed
0
)(
log
1
log
1
a
Laaak nn  

.
Taking 1k in Cor. 1 , we get the following result:
Corollary 3: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n such that for some 10,  n and for
some 1 o ,
 aaaa nn   11 ......
and
001211 ...... aaaaaaaL   
Then the number f zeros of P(z) in 10,  z does not exceed
0
)1(
log
1
log
1
a
aLaaa nn  


.
Taking 1 in Theorem 1 , we get the following result:
Corollary 4: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n with ,)Im(,)Re( jjjj aa  
nj ,......,2,1,0 such that for some 10,  n and for some ,1k ,
    11 ......nnk
and
001211 ......    L .
Then the number f zeros of P(z) in 10,  z does not exceed
0
0
2)1(
log
1
log
1
a
Lkka
n
j
jnnn 
 


.
Taking 1k in Theorem 1 , we get the following result:

3. On The Zeros Of A Polynomial Inside The Unit Disc
69
Corollary 5: Let 

n
j
j
j zazP
0
)( be a polynomial of degree n with ,)Im(,)Re( jjjj aa  
nj ,......,2,1,0 such that for some 10,  n and for some 1 o ,
    11 ......nn
and
001211 ......    L .
Then the number f zeros of P(z) in 10,  z does not exceed
0
0
2)1(
log
1
log
1
a
La
n
j
jnn 
 


.
Similarly for other different values of the parameters, we get many other interesting results.
III. LEMMA
For the proof of Theorem 1, we need the following result:
Lemma: Let f (z) be analytic for 0)0(,1  fz and Mzf )( for 1z . Then the number of zeros of
f(z) in 10,  z does not exceed
)0(
log
1
log
1
f
M

.
(for reference see [6] ).
IV. PROOF OF THEOREM 1
Consider the polynomial
)()1()( zPzzF 



 zaazaazaaza
azazazaz
n
nn
n
n
n
n
n
n
)()(......)(
)......)(1(
1
1
11
1
01
1
1








001 )(...... azaa 
1
1
1
211
1
)(......)()()1( 




 
  zzzkzkza n
nn
n
nn
n
n
n
n
})(......
){()(......)()1(
001
10011
1

 




 

z
zizzz n
nn
For 1z , we have, by using the hypothesis
  )1(......)1()( 1211   nnnnnn kkazF
001
2110011 ..............

 

  nnnn
  )1(......)1( 1211   nnnnnn kka
001
2110011 ..............

 

  nnnn


n
j
jnnn Lkka
0
2)()1(  
Since F(z) is analytic for 1z , 0)0( 0  aF , it follows by the Lemma that the number of zeros

4. On The Zeros Of A Polynomial Inside The Unit Disc
70
of F(z) in 10,  z des not exceed
0
0
2)()1(
log
1
log
1
a
Lkka
n
j
jnnn 
 


.
Since the zeros of P(z) are also the zeros of F(z) , it follows that the number of zeros
of P(z) in 10,  z des not exceed
0
0
2)()1(
log
1
log
1
a
Lkka
n
j
jnnn 
 


.
That completes the proof of Theorem 1.
REFERENCES
[1]. 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.
[2]. K.K.Dewan, Theory of Polynomials and Applications, Deep & Deep Publications, 2007,
Chapter 17.
[3]. M. H. Gulzar, On the Number of Zeros of a Polynomial in a Prescribed Region, Reseach Journal of
Pure Algebra, Vol.2(2) , 2012, ,35-46.
[4]. M. Marden, Geometry of Polynomials, Math. Surveys No. 3, Amer. Math.Soc.(1966).
[5]. Q. G Mohammad, On the Zeros of Polynomials, Amer. Math. Monthly, Vol.72 , 1965, 631-633
[6]. E. C. Titchmarsh, Theory of Functions, 2nd
Edition, Oxford University Press, 1949.