This document summarizes several theorems regarding the location of zeros of polynomials: Theorem 1 generalizes previous results (Theorems C and E) by proving bounds on the location of zeros of polynomials of the form P(z) = a0 + a1z + ... + aμzμ + zn, where 0 ≤ μ ≤ n-1. Theorem 2 further generalizes Theorem E by providing bounds on the location of zeros of polynomials of the same form as in Theorem 1, under the additional condition that 0 < aj-1 ≤ kaj, where k > 0. The proofs of Theorems 1 and 2 apply Holder's inequality and results from previous theorems

1. I nternational Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 7
Location Of The Zeros Of Polynomials
M.H. Gulzar
Depart ment of Mathemat ics
University of Kashmir, Srinagar 190006.
Abstract: In this paper we prove some results on the location of zeros of a certain class of polynomials which among other
things generalize some known results in the theory of the distribution of zeros of polynomials.
Mathematics Subject Cl assification: 30C10, 30C15
Keywords and Phrases : Polynomial, Zeros, Bounds
1. Introduction And Statement Of Results
A celebrated result on the bounds for the zeros of a polynomial with real coefficients is the follo wing theorem ,known
as Enestrom –Kakeya Thyeorem[1,p.106]
Theorem A: If 0  a 0  a1  ......  a n , then all the zeros of the polynomial
P( z)  a0  a1 z  a 2 z 2  ......  a n1 z n1  a n z n
lie in z 1 .
Regarding the bounds for the zeros of a polyno mial with leading coefficient unity, Montel and Marty [1,p.107] proved the
following theorem:
Theorem B : All the zeros of the polynomial
P( z)  a0  a1 z  a 2 z 2  ......  a n1 z n1  z n
1
lie in z  max( L, L ) where L is the length of the polygonal line jo ining in succession the points
n
0, a0 , a1 ,......, a n1,1 ; i.e.
L  a0  a1  a0  ......  a n1  a n2  1  a n1 .
Q .G. Mohammad [2] proved the following generalizat ion of Theorem B:
Theorem C: All the zeros of the polynomial 0f Theorem A lie in
1
z  R  max( L p , L p n )
where
1 n 1 1
L p  n ( a j ) , p 1  q 1  1 .
q p p
j 0
The bound in Theorem C is sharp and the limit is attained by
1 n 1
P( z )  z n  ( z  z n  2  ......  z  1) .
n
Letting q  ∞ in Theorem C, we get the following result:
1
Theorem D: All the zeros of P(z) 0f Theorem A lie in z  max( L1 , L1 n ) where
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2. I nternational Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 7
n 1
L1   a i ) .
i 0
Applying Theorem D to the polynomial (1-z)P(z) , we get Theorem B.
Q .G. Mohammad , in the same paper , applied Theorem D to prove the follo wing result:
Theorem E: If 0  a j 1  ka j , k  0 , then all the zeros of
P( z)  a0  a1 z  a 2 z 2  ......  a n1 z n1  a n z n
1
lie in z  max( M , M ) where n
(a 0  a1  ......  a n 1 )
M (k  1)  k .
an
The aim o f this paper is to give generalizations of Theorems C and E. In fact, we are going to prove the follo wing results:
Theorem 1: All the zeros of the polynomial
P( z)  a0  a1 z  a 2 z 2  ......  a  z   z n ,0    n  1
lie in
1
z  R  max( L p , L p n )
where
1  1
L p  n ( a j ) , p 1  q 1  1 .
q p p
j 0
Remark 1: Taking  =n-1, Theorem 1 reduces to Theorem C.
Theorem 2: If 0  a j 1  ka j , k  0 , then all the zeros of
P( z)  a0  a1 z  a 2 z 2  ......  a  z   a n z n ,0    n  1 ,
1
lie in z  max( M , M ) where n
(a 0  a1  ......  a  )
M (k  1)  k .
an
Remark 2: Taking  =n-1 , Theorem 2 reduces to Theorem E and taking  =n-1 , k=1, Theorem 2 reduces to Theorem A
due to Enestrom and Kakeya..
2. Proofs Of Theorems
Proof of Theorem 1 . Applying Ho lder’s inequality, we have
P( z )  a0  a1 z  a 2 z 2  ......  a  z   z n
  1 1 
 z 1   a j 1 n  j 1 
n

 j 1 z 

 1  1
1
1

1  n q ( a j 1
p
 z ) p .
n
( n  j 1) p

 j 1 z 

1
1 1
If L p  1, max( L p , L p n )  L p . Let z  1 . Then ( n  j 1) p
 p , j  1,2,......,   1 .
z z
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3. I nternational Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 7
Hence it follows that for z  Lp ,
 1
1 
n nq  p p  n Lp 
P( z )  z 1  ( a j )   z 1    0.

z j 0
 
 z 

 
1 1
Again if L p  1, max( L p , L p n )  L p n . Let z  1 . Then
1 1
( n  j 1) p
 np
, j  1,2,......,   1 .
z z
1
Hence it follows that for z  Lp n ,
 1
1 
n nq  p p   Lp 
P( z )  z 1  n ( a j )   z 1  n   0.
n
 z j 0  
 z 
 
1
Thus P(z) does not vanish for z  max( L p , L p n ) and hence the theorem fo llo ws.
Proof of Theorem 2. Consider the polynomial
F ( z)  (k  z) P( z)  (k  z)(a0  a1 z  ......  a  z   a n z n )
 ka0  (ka1  a0 ) z  (ka2  a1 ) z 2  ......  (ka  a  1 ) z   a  z  1
 kan z n  a n z n 1
F ( z)
Applying Theorem C to the polynomial , we find that
an
k (a 0  a1  ......  a  )  (a 0  a1  ......  a  1  a  )  kan
L1 
an
(k  1)(a 0  a1  ......  a  )
 k
an
=M
and the theorem follows.
References
[1] M. Marden , The Geo metry of Zeros,, A mer.Math.Soc.Math.Surveys ,No.3 ,New
York 1949.
[2] Q.G. Mohammad , Location of the Zeros of Po lynomials , A mer. Math. Monthly ,
vol.74,No.3, March 1967.
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