ISSN (e): 2250 – 3005 || Volume, 07 || Issue, 02|| February – 2017 || International Journal of Computational Engineering Research (IJCER)
Location of Regions Containing all or Some Zeros of a Polynomial M.H. Gulzar Department of Mathematics, University of Kashmir, Srinagar
ABSTRACT In this paper we locate the regions which contain all or some of the zeros of a polynomial when the coefficients of the polynomial are restricted to certain conditions. Mathematics Subject Classification: 30C10, 30C15. Keywords and Phrases: Coefficients, Polynomial, Regions, Zeros.
I. INTRODUCTION The following result known as the Enestrom-Kakeya Theorem [3] is of fundamental importance on locating a region containing all the zeros of a polynomial with monotonically increasing positive coefficients: n
Theorem A: Let P ( z )
a jz
j
be a polynomial of degree n such that
j0
a n a n 1 ...... a 1 a 0 0 .
Then all the zeros of P(z) lie in z 1 . In the literature several generalizations and extensions of this result are available []. Recently Gulzar et al [1,2] proved the following results: n
Theorem B: Let P ( z )
a jz
j
be a polynomial of degree n with
j0
Re( a j ) j , Im( a j ) j , j 0 ,1, 2 ,......, n
such that for some , 0 n 1 and for some
k 1, o 1 ,
k
n
n 1
......
and L
1
1
2
...... 1
0
0
.
Then all the zeros of P(z) lie in n
z
( k 1 )
k n
n
(1 )
L 2 j0
an
j
.
an n
Theorem C: Let P ( z )
a jz
j
be a polynomial of degree n with
j0
Re( a j ) j , Im( a j ) j , j 0 ,1, 2 ,......, n
such that for some , 0 n 1 and for some
k 1, o 1 ,
k
n
n 1
......
1
and L
1
1
2
...... 1
0
0
.
Then the number f zeros of P(z) in z , 0 1 does not exceed
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Location of Regions Containing all or Some Zeros of a Polynomial n
a n ( k 1) 1 1
log
k
n
n
L (1 )
2
j
j0
log
.
a0
II. MAIN RESULTS In this paper we prove the following generalizations of Theorems B and C: n
Theorem 1: Let P ( z )
a jz
j
be a polynomial of degree n with
j0
Re( a j ) j , Im( a j ) j , j 0 ,1, 2 ,......, n
such that for some , 0 n 1 and for some
k 1 , k 2 1, o 1 ,
k 1
k 2
n
n 1
......
and L
1
1
2
...... 1
0
0
.
Then all the zeros of P(z) lie in n
z
( k 1 1 )
n
k 1
( k 2 1 )
an
1
n
( k 2 1)
n 1
L (
) 2
j
j0
an
.
an
n
Theorem 2: Let P ( z )
a jz
j
be a polynomial of degree n with
j0
Re( a j ) j , Im( a j ) j , j 0 ,1, 2 ,......, n
such that for some , 0 n 1 and for some
k 1 , k 2 1, o 1 ,
k 1
n
k 2
n 1
......
...... 1
0
1
and L
1
1
2
0
.
Then the number of zeros of P(z) in z , 0 1 does not exceed 1 log
1
1
log
a0
n
[ a n ( k 1 1)
n
k 1
n
2 ( k 2 1)
n 1
L (
) 2
j
.
j0
Remark 1: Taking k 2 1 , Theorem 1 reduces to Theorem B and Theorem 2 reduces to Theorem C. For different values of the parameters in Theorems 1 and 2 , we get many interesting results. For example taking 1 , Theorem 1 gives the following result: n
Corollary 1: Let P ( z )
a jz
j
be a polynomial of degree n with
j0
Re( a j ) j , Im( a j ) j , j 0 ,1, 2 ,......, n
such that for some , 0 n 1 and for some
k 1 , k 2 1, ,
k 1
n
k 2
n 1
......
and L
1
1
2
...... 1
0
0
.
Then all the zeros of P(z) lie in
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Location of Regions Containing all or Some Zeros of a Polynomial n
k 1
( k 1 1 )
z
( k 2 1 )(
n
n 1
n 1
) L
2
j
j0
n
an
.
an
Taking 1 , Theorem 2 gives the following result: n
Corollary 2: Let P ( z )
j
a jz
be a polynomial of degree n with
j0
Re( a j ) j , Im( a j ) j , j 0 ,1, 2 ,......, n
such that for some , 0 n 1 and for some
k 1 , k 2 1, ,
k 1
k 2
n
......
n 1
1
and L
1
1
...... 1
2
0
0
.
Then the number f zeros of P(z) in z , 0 1 does not exceed 1 log
1
1
log
a0
n
[ a n ( k 1 1)
k 1
n
n
2 ( k 2 1)
L
n 1
2
j
].
j0
III. LEMMA For the proof of Theorem 2, we need the following result: Lemma: Let f (z) be analytic for z 1 , f ( 0 ) 0 and f ( z ) M for z 1 . Then the number of zeros of 1
f(z) in z , 0 1 does not exceed
log
M
log
1
.
f (0 )
(for reference see [4] ). 3. Proofs of Theorems Proof of Theorem 1: Consider the polynomial F ( z ) (1 z ) P ( z )
(1 z )( a n z anz
n 1
n
a n 1 z
n 1
( a n a n 1 ) z
...... a 1 z a 0 ) n
...... ( a 1 a ) z
1
( a a 1 ) z
...... ( a 1 a 0 ) z a 0 an z
n 1
( k 1 1 ) n z
( k 2 1 ) (
n 1
1
z
)z
n 1
n
( k 1
(
n2
n
k 2
n3
)z
n 1
n2
)z
( k 2 1 )
n
...... (
...... ( 1 0 ) z
i {(
0
1
z
n
n 1
n 1
n
)z
( k 2
)z n
1
n 1
)z
n2
( 1 )
z
n 1
1
......
( 1 0 )z 0}
For z 1 so that
1 z
j
1 , j 1 , 2 ,......, n , we have, by using the hypothesis
F ( z ) a n z ( k 1 1 )
(
n2
n
( k 2 1 )
n3
) z.
...... 1
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0
n2
n 1
z
n
[ k 1
.....
z
0
{
n
1
n
n 1
k 2
z
z n
n 1
1
z
n
k2 1
1
z
n 1
1
z
n 1
k 2
1
n 1
z
...... 1 0 z 0 ]
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n2
.z
n 1
Location of Regions Containing all or Some Zeros of a Polynomial n
z [ a n z ( k 1 1 )
n2
z
n3
1
1 0
0
n
1
k 2
n 1
n
( k 2 1)
n 1
n 1
z
n 1
k 2
n 1 n 2
n2
z
z
n 1
z
(1 )
n n 1
( k 2 1)
n
n
1
n
......
z
}]
n
z
n 1
0
z
0
z [ a n z ( k 1 1 )
n 1
n 1
k 2
1
z
z
z
{ k 1
n 1
......
2
......
( k 2 1 )
n
n2
n2
n 1
{ k 1
n3
.....
...... 1
0
0
n
k 2 1
n 1
( k 2 1)
n 1
(1 )
n n 1 n 1 n 2 ......
1 0 0 }] n
z [ a n z ( k 1 1 ) k 2
n 1
1
n
( k 2 1)
n2
n2
n3
...... 1
0
n 1
{ k 1
.....
0
n
k 2
1
n 1
( k 2 1)
(1 )
n 1
n n 1 n 1 n 2 ......
1 0 0 }] n
z [ a n z ( k 1 1 )
n
( k 2 1 )
n 1
{ k 1
n
( k 2 1)
n 1
L
n
(
) 2 j }] j0
0
if n
a n z ( k 1 1 )
n
( k 2 1 )
n 1
k 1
n
( k 2 1)
n 1
L (
) 2
j
j0
i.e. if n
z
( k 1 1 )
n
( k 2 1 )
an
k 1 1
n
( k 2 1)
n 1
L (
) 2
j
j0
an
.
an
This shows that those zeros of F(z) whose modulus is greater than 1 lie in n
z
( k 1 1 )
n
( k 2 1 )
an
k 1 1
n
( k 2 1)
n 1
L (
) 2 j0
an
j
.
an
Since the zeros of F(z) whose modulus is less than or equal to 1 already satisfy the above inequality and since the zeros of P(z) are also the zeros of F(z) , it follows that all the zeros of P(z) lie in n
z
( k 1 1 ) an
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n
( k 2 1 )
k 1 1
n
( k 2 1)
n 1
L (
) 2 j0
an
j
.
an
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Location of Regions Containing all or Some Zeros of a Polynomial That proves Theorem 1. Proof of Theorem 2: Consider the polynomial F ( z ) (1 z ) P ( z )
(1 z )( a n z anz
n 1
n
n 1
a n 1 z
( a n a n 1 ) z
...... a 1 z a 0 ) n
...... ( a 1 a ) z
1
( a a 1 ) z
...... ( a 1 a 0 ) z a 0 n 1
an z
( k 1 1 ) n z
( k 2 1 ) (
n 1
1
z
n 1
)z
n
( k 1
(
n2
k 2
n
n3
)z
n 1
n2
( k 2 1 )
n
)z
...... (
...... ( 1 0 ) z
i {(
0
n 1
n 1
( k 2
n
z
1
n
n 1
)z
)z
1
n 1
n2
( 1 )
)z
z
n 1
1
......
n
( 1 0 )z 0}
For z 1 , we have, by using the hypothesis F ( z ) a n ( k 1 1)
n2
k 1
n
n3
n
k 2
......
1
n 1
( k 2 1)
n 1
(1 )
k 2
( k 2 1)
n2
1
n 1
...... 1
0
0
n n 1 n 1 n 2 ....... 1 0 0 a n ( k 1 1)
n2
n3
n 1
n 1
......
n
k 1
1
n
k 2
n 1
( k 2 1)
(1 )
n 1
k 2
1
n 1
n2
( k 2 1)
...... 1
0
0
n 1
n
n 2 ....... 1 0 0
a n ( k 1 1)
n
k 1
n
2 ( k 2 1)
n 1
)
L (
)
n
2
j
j0
Since F(z) is analytic for z 1 , F ( 0 ) a 0 0 , it follows by the Lemma that the number of zeros of F(z) in z , 0 1 des not exceed 1 log
1
log
1 a0
n
[ a n ( k 1 1)
n
k 1
n
2 ( k 2 1)
n 1
L (
) 2
j
.
j0
Since the zeros of P(z) are also the zeros of F(z) , it follows that the number of zeros of P(z) in z , 0 1 des not exceed 1 log
1
log
1 a0
n
[ a
n
( k 1 1)
n
k 1
n
2(k
2
1)
n 1
)
L (
) 2
j
.
j0
That completes the proof of Theorem 2.
REFERENCES [1]. [2]. [3]. [4].
M. H. Gulzar, B.A. Zargar and A. W. Manzoor, On the Zeros of a Family of Polynmials, Int. Journal Of Engineering Technology and Management Vol. 4, Issue 1 (Jan-Feb 2017), 07-10. M. H. Gulzar, B.A. Zargar and A. W. Manzoor, On the Zeros of a Polynomial inside the Unit Disc, Int. Journal of Engineering Research and Development, Vol.13, Issue 2 (Feb.2017), 17-22. M. Marden, Geometry of Polynomials, Math. Surveys No. 3, Amer. Math.Soc.(1966). E. C. Titchmarsh, Theory of Functions, 2nd Edition Oxford University Press, 1949.
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