The document summarizes previous research on bounding the number of zeros of polynomials in certain regions. It presents four theorems (Theorems A-D) from previous works that provide upper bounds on the number of zeros inside circles or annuli based on the coefficients of the polynomials. The paper then proves a new Theorem 1, which establishes that a polynomial P(z) with certain coefficient properties has no zeros in the region z < a0/M1 for R ≥ 1 or z < a0/M2 for R ≤ 1, where M1 and M2 are defined in terms of the polynomial coefficients. Combining this with Theorem D gives a new Theorem 2, which provides upper bounds on the number of zeros inside rings

1. International Journal of Computational Engineering Research||Vol, 03||Issue, 11||
Polynomials having no Zero in a Given Region
1
M. H. Gulzar
1
Department of Mathematics University of Kashmir, Srinagar 190006
ABSTRACT:
In this paper we consider some polynomials having no zeros in a given region. Our results when
combined with some known results give ring –shaped regions containing a specific number of zeros of
the polynomial.
Mathematics Subject Classification: 30C10, 30C15
Keywords and phrases: Coefficient, Polynomial, Zero.
I.
INTRODUCTION AND STATEMENT OF RESULTS
In the literature we find a large number of published research papers concerning the number of zeros
of a polynomial in a given circle. For the class of polynomials with real coefficients, Q. G. Mohammad [5]
proved the following result:

Theorem A: Let
P( z )   a j z j be a polynomial o f degree n such that
j 0
an  an1  ......  a1  a0  0 .
1
Then the number of zeros of P(z) in z 
does not exceed
2
a
1
log n .
log 2
a0
Bidkham an d Dewan [1] generalized Theorem A in the following way:
1

Theorem B: Let
P( z )   a j z j be a polynomial o f degree n such that
j 0
a n  a n1  ......  a k 1  a k  a k 1  ......  a1  a0  0 ,
1
for some k ,0  k  n .Then the number of zeros of P(z) in z 
does not exceed
2
 a  a0  a n  a0  2 a k
1

log  n
log 2
a0


.




Ebadian et al [2] generalized the above results by proving the following
results:

Theorem C: Let
P( z )   a j z j be a polynomial o f degree n such that
j 0
a n  a n1  ......  a k 1  a k  a k 1  ......  a0
for some k, 0  k  n .Then the number of zeros of P(z) in
z 
R
,R>0, does not exceed
2
 a n R n 1  a 0  R k (a k  a 0 )  R n (a k  a n ) 
1

 for R  1
log 

log 2
a0




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2. Polynomials having no Zero in a Given Region
And
 a n R n 1  a 0  R( a k  a 0 )  R n ( a k  a n ) 
1


log 

log 2
a0




for
R 1 .
M.H.Gulzar [3] generalized the above result by proving the following result:

Theorem D: Let
P( z )   a j z j be a polynomial o f degree n with Re( a j )   j , Im(a j )   j such that
j 0
k , ,  ,0  k  1,0    1, 0    n ,
k n   n1  ......    1       1  ......   0 .
R
Then the number of zeros of P(z) in z  ( R  0, c  1) does not exceed
c
for some
 1

n 1
 a 0  R  [    (  0   0 )   0   0     2  j
 an R
j 1

n 1

n
 R [ n  k( n   n )       n  2   j ]
1

j   1
log 
log c
a0






for
R 1

]











and
 1

a n R n 1  a 0  R[    (  0   0 )   0   0     2  j

j 1

n 1

n
 R [ n  k( n   n )       n  2   j ]
1

j   1
log 
log c
a0







]











In this paper we prove the following result:

Theorem 1: Let
P( z )   a j z j be a polynomial o f degree n with Re( a j )   j , Im(a j )   j such that
j 0
for some
k , ,  ,0  k  1,0    1, 0    n ,
k n   n1  ......    1       1  ......   0 .
Then P(z) has no zero in
z
a0
M1
for
R  1 and no zero in z 
a0
for
M2
R 1
where
n
M 1  an R n1  R n [  n  k (  n   n )    ]      n  2   j ]
j   1
 1
 R  [    (  0   0 )   0   0     2  j ]
j 1
and
n
M 2  an R n1  R n [  n  k (  n   n )    ]      n  2   j ]
j   1
 1
 R[    (  0   0 )   0   0     2  j ] .
j 1
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3. Polynomials having no Zero in a Given Region
Combining Theorem 1 with Theorem D, we get the following result:

Theorem 2: Let
P( z )   a j z j be a polynomial o f degree n with Re( a j )   j , Im(a j )   j such that
j 0
for some
k , ,  ,0  k  1,0    1, 0    n ,
k n   n1  ......    1       1  ......   0 .
a0
Then the number of zeros of P(z) in
M1
 z
R
( R  0, c  1) does not exceed
c
 1

a n R n 1  a 0  R  [    (  0   0 )   0   0     2  j

j 1

n 1

n
 R [ n  k( n   n )       n  2   j ]
1

j   1
log 
log c
a0






for
and the number of zeros of P(z) in
a0
M2
 z
R 1

]











R
( R  0, c  1) does not exceed
c
 1

n 1
 a 0  R[    (  0   0 )   0   0     2  j
 an R
j 1

n 1

n
 R [ n  k( n   n )       n  2   j ]
1

j   1
log 
log c
a0






for
R 1 ,

]











where M 1 and M 2 are as given in Theorem 1.
For different values of the parameters, we get many interesting results including some already known
results.
2. Proofs of Theorems
Proof of Theorem 1: Consider the polynomial
F(z) =(1-z)P(z)
 (1  z )(a n z n  a n1 z n1  ......  a1 z  a0 )
 a n z n1  (a n  a n1 ) z n  ......  (a1  a0 ) z  a0
 a n z n 1  a0  [(k n   n 1 )  (k  1) n ]z n 
n 1
 (

j  1

j
  ( j   j 1 ) z j  [( 1   0 )  (  1) 0 ]z  i{
j 2
  j 1 ) z j
n
 (

j  1
j
  j 1 ) z j

  (  j   j 1 ) z j }
j 1
 a 0  G( z ) , where
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4. Polynomials having no Zero in a Given Region
G( z )  a n z n 1  a0  [(k n   n 1 )  (k  1) n ]z n 
n 1
 (

j  1


j 2
  j 1 ) z j
j
j 1
  ( j   j 1 ) z j  [( 1   0 )  (  1) 0 ]z  i  (  j   j 1 ) z j
z  R , we have, by using the hypothesis
For
G( z )  a n R n 1  [( n 1  k n )  (1  k )  n ]R n 
n 1
 (

j  1
j 1
  j )R j

n
j 1
j 1
  ( j   j 1 ) R j  [( 1   0 )  (1   )  0 ]R   (  j   j 1 ) R j
which gives
n
G( z )  an R n1  R n [  n  k (  n   n )    ]      n  2   j ]
j   1
 1
 R  [    (  0   0 )   0   0     2  j ]
j 1
 M1
for
R 1
and
n
G( z )  an R n1  R n [  n  k (  n   n )    ]      n  2   j ]
j   1
 1
 R[    (  0   0 )   0   0     2  j ]
j 1
 M2
R  1.
Since G(z) is analytic in z  R and G(0)=0, it follows by Schwarz Lemma that
for
G( z )  M 1 z for R  1and G( z )  M 2 z for R  1.
Hence, for R  1 ,
F ( z )  a0  G( z )
 a0  G( z )
 a0  M 1 z
0
if
z 
a0
.
M1
R  1,
F ( z )  a0  G( z )
And for
 a0  G( z )
 a0  M 2 z
0
if
z 
a0
.
M2
This shows that F(z) has no zero in
z
a0
M1
for
R  1 and no zero in z 
a0
M2
for
R  1 . But the zeros of
P(z) are also the zeros of F(z). Therefore, the result follows.
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5. Polynomials having no Zero in a Given Region
REFERENCES
[1]
[2]
[3]
[4]
[5]
M.Bidkham and K.K. Dewan, On the zeros of a polynomial, Numerical Methods and Approximation Theory,III , Nis (1987), 121128.
A.Ebadian, M.Bidkham and M.Eshaghi Gordji, Number of zeros of a polynomial in a given domain,Tamkang Journal of
Mathematics, Vol.42, No.4,531-536, Winter 2011.
M.H.Gulzar, On the Number of Zeros of a Polynomial in a Given Domain, International Journal of Scientific and Research
Publications, Vol.3, Issue 10, October 2013.
M.Marden , Geometry of Polynomials , Mathematical surveys, Amer. Math. Soc. , Rhode Island , 3 (1966).
Q.G.Mohammad, Location of the zeros of polynomials, Amer. Math. Monthly , 74 (1967), 290-292.
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