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 n1 z n1 a n z n
z 1 .
lie in
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 n1 z n1 z n lie in
1 n
z max( L, L ) where L is the length of the polygonal line jo ining in succession the points
0, a0 , a1 ,......, a n1,1 ; i.e.
L a0 a1 a0 ...... a n1 a n2 1 a n1 . 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 q
n 1
1 p
L p n ( a j ) , p 1 q 1 1 . p
j 0
The bound in Theorem C is sharp and the limit is attained by
P( z ) z n Letting
1 n 1 ( z z n 2 ...... z 1) . n
q ∞ in Theorem C, we get the following result: 1
Theorem D: All the zeros of P(z) 0f Theorem A lie in
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z max( L1 , L1 n ) where
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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 n1 z n1 a n z n lie in
1 n
z max( M , M ) where M
(a 0 a1 ...... a n 1 ) (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 q
1 p
L p n ( a j ) , p 1 q 1 1 . p
j 0
=n-1, Theorem 1 reduces to Theorem C. 0 a j 1 ka j , k 0 , then all the zeros of
Remark 1: Taking Theorem 2: If
P( z) a0 a1 z a 2 z 2 ...... a z a n z n ,0 n 1 , lie in
1 n
z max( M , M ) where M
(a 0 a1 ...... a ) an
(k 1) k .
Remark 2: Taking =n-1 , Theorem 2 reduces to Theorem E and taking due to Enestrom and Kakeya..
=n-1 ,
k=1, Theorem 2 reduces to Theorem A
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 j 1 z 1 1 1 p 1 n q p . z 1 n ( a j 1 ) ( n j 1) p j 1 z 1 1 1 p , j 1,2,......, 1 . If L p 1, max( L p , L p n ) L p . Let z 1 . Then ( n j 1) p z z n
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I nternational Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 7
Hence it follows that for
z Lp ,
1 1 Lp p p nq n n P( z ) z 1 ( a j ) z 1 0. z j 0 z
1
Again if
1
L p 1, max( L p , L p n ) L p n . Let z 1 . Then 1 z
( n j 1) p
1 z
np
, j 1,2,......, 1 . 1
Hence it follows that for
z Lp n ,
1 1 q Lp p p n n n P( z ) z 1 n ( a j ) z 1 n 0. 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) , we find that an k (a 0 a1 ...... a ) (a 0 a1 ...... a 1 a ) kan
Applying Theorem C to the polynomial
L1
an (k 1)(a 0 a1 ...... a ) an
k
=M and the theorem follows.
References [1] [2]
M. Marden , The Geo metry of Zeros,, A mer.Math.Soc.Math.Surveys ,No.3 ,New York 1949. Q.G. Mohammad , Location of the Zeros of Po lynomials , A mer. Math. Monthly , vol.74,No.3, March 1967.
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