Applied Mathematics, 2011, 2, 1356-1358
doi:10.4236/am.2011.211189 Published Online November 2011 (http://www.SciRP.org/journal/am)
On the Zeros of a Polynomial
Mohammad Syed Pukhta
Division of Agricultural Engineering, Sher-e-Kashmir University of Agricultural Sc iences
& Technology of Kashmir, Srinagar, India
Received February 27, 2011; revised May 25, 2011; accepted J une 3, 2011
Abstract
In this paper we consider the problem of finding the estimate of maximum number of zeros in a prescribed
region and the results which we obtain generalizes and improves upon some well known results.
Keywords: Polynomial, Zeros, Complex Number, Prescribed Region
1. Introduction
Let be a polynomial of degree n such
that

0
ni
i
i
pz az


110
0
nn
aa aa


then according to a well known result of Enstrom and
Kakeya, the polynomial


,pz does not vanish in
1.z concerning the number of zeros of the polyno-
mial in the region 1,
2
z the following result is due to
Mohammad [1].
Theorem A. Let be a polynomial of
degree n such that

0
ni
i
i
pz az


110
0,
nn
aa aa


then the number of zeros of


pz in 1,
2
z does not
exceed
1
1log
log 2
n
o
a
a
.
Dewan [2] generalized Theorem A to the polynomials
with complex coefficients and obtained the following
result.
Theorem B. If is a polynomial of de-

0
ni
i
i
pz az


gree n with complex coefficients such that
π
arg,0,1, 2,,
2
i
ai  
11
,
nn
aa aa

0
then the number of zeros of in

pz 1
2
z does not
exceed

1
0
0
cossin12sin
1log
log 2
n
ni
i
aa
a
 


 
.
Theorem C. Let be a polynomial of

0
ni
i
i
pz az


degree n with complex coefficients. If
Re, Im,
ii ii
aa



 for and 0,1, ,in
110
0,
nn

  
 then the number of zeros
of


pz in 1
2
z

does not exceed
0
0
1
1log
log 2
n
ni
i
a




 .
In this paper we generalize Theorem B and Theorem C
under less restrictive conditions on the coefficients,
which also improve upon them. More precisely, we
prove the following.
Theorem 1. Let be a polynomial of

0
ni
i
i
pz az


degree n with complex coefficients, such that
π
arg,0,1,2,,.
2
i
ai  n for some real

and
110
0
nn
aa aa


n for some real

and
then the number of zeros of in

pz ,z

 does not
exceed

M. S. PUKHTA1357

1
0
0
cossin12sin
1log
1
log
n
ni
a
i
a
a
 




 


.
where 01


Theorem 2. Let be a polynomial of

0
ni
i
i
pz az


degree n with complex coefficients. If
Re, Im,
ii ii
aa

  for 0,1,,in


0
and
11nn 0,



  0
n

, then the number of
zeros of in

pz ,0z  1   does not exceed
0
0
1
1log
1
log
n
ni
i
a






.
2. Lemma
We need the following lemma for proof of the theorems.
Lemma. Let be a polynomial of de-
gree n such that

0
ni
i
i
pz az


1
π
arg ;
2
ii
aa 


i
a for some 0,1,2,,,in
then

11 1
cos sin
iii ii i
aaa aa a


 
 .
The proof of the above lemma is omitted as it follows
from the lemma in [3].
3. Proof of the Theorems
Proof of Theorem 1. Consider the polynomial
 




1
11
1
1
1
12100
11






 
 
 


nn
nn
nn
nnn
n
nn
0
g
zzpzzazaz aza
azaaz
aaz aaza
For 1z, we have

 

If


f
z is regular, and

00f

f
zM in
1

, then ([4], p.171) the number of zeros of


f
z z




10
1
11
11
1
0
0
1
0
cossin 0
b
y using Lemma
cossin1 2sin
cossin 1
cossin1 2sin.
n
nii
i
nn
niiii
ii
n
ni
i
n
ni
i
gzaaaa
aaa aa
aa
a
aa

 

 








 
  










a 
in ,0 1z 

 does not exceed

1log
10
log
M
.
f

Apply this result to


g
z in z

 does not exceed

1
0
0
cossin1 2sin
1log
1
log
n
ni
i
aa
a
 




 


.
All the number of zeros of in
()pz z

 is also
equal to the number of zeros of ()
g
z in z


. This
completes proof of Theorem 1.
Proof of Theorem 2.
Consider
 
1
10
1
1
n
ni
nii
i
g
zzpzaz aaz



a



For 1z

,




10
1
110
1
11
11
0
2
n
nii
i
n
n niiii
i
nn
nniii i
ii
n
ni
i
gzaa aa
0
0












 
 
 









and using the same argument as in proof of Theorem 1,
the proof of Theorem 2 follows.
Remark 1. For 1,
2


Theorem 1 is a refinement of
Theorem B and for 1,
2


and 0,

  it gives a
refinement of Theorem A.
Remark 2. Theorem C can be deduced as a particular
case of Theorem 2 by putting 1
2
 . If we put
0, 0
iin


 in Theorem 2, we can deduce Theorem
A.
Corollary 1. Let be a polynomial of
degree n, such that

0
ni
i
i
pz az


11
,
nn 0



 
then the number of zeros of p(z) in ,01,z  
does not exceed
0
1
1log
1
log
n
a


.

M. S. PUKHTA 1358
4. Acknowledgements
Author is highly thankful to the referees for their valu-
able suggestions.
5. References
[1] N. K. Govil and Q. I. Rehman, “On the Enstrom Kakeya
Theorem,” Tohoku Mathematical Journal, Vol. 20, No. 2,
1968, pp. 126-136. doi:10.2748/tmj/1178243172
[2] Q. G. Mohammad, “On the Zeros of the Polynomials,”
American Mathematical Monthly, Vol. 72, No. 6, 1965,
pp. 631-633. doi:10.2307/2313853
[3] E. C. Titchmarsh, “The Theory of Functions,” 2nd Edi-
tion, Oxford University Press, London, 1939.
[4] K. K. Dewan, “Extremal Properties and Coefficient Es-
timates for Polynomials with Restricted Zeros and on
Location of Zeros of Polynomials,” Ph.D Thesis, Indian
Institutes of Technology, Delhi, 1980.