AJ02702120214

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

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 n1 z n1  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 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 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 n1 z n1  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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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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