ISSN (e): 2250 – 3005 || Volume, 07 || Issue, 03|| March – 2017 || International Journal of Computational Engineering Research (IJCER)
Location of Zeros of Polynomials M.H. Gulzar , B.A.Zargar , A.W.Manzoor 1
2
3
Department of Mathematics, University of Kashmir, Srinagar
ABSTRACT In this paper, restricting the coefficients of a polynomial to certain conditions, we locate a region containing all of its zeros. Our results generalize many known results in addition to some interesting results which can be obtained by choosing certain values of the parameters. Mathematics Subject Classification: 30C10, 30C15. Key Words and Phrases: Coefficients, Polynomial, Zeros.
I. INTRODUCTION On the location of zeros of a polynomial with real coefficients Enestrom and Kakeya proved the following theorem named after them as the Enestrom - Kakeya Theorem [9,10] : n
Theorem A: all the zeros of a polynomial P ( z )
a jz
j
satisfying a n a n 1 ....... a 1 a 0 0 lie
j0
in z 1 . The above theorem has been generalized in different ways by various authors [2-11]. Recently Gulzar et al [7] proved the following result n
Theorem
B:
P(z)
Let
a jz
j
be
a
polynomial
of
degree
with
n
j0
Re( a j ) j , Im( a j ) j , j 0 ,1 , 2 ,......,
n
n 1
n such that for some , , 0 n 1, 0 n 1 ,
...... ,
n n 1 ...... ,
and L 1 1 2 ...... 1 0 0 , M 1 1 2 ...... 1 0 0 .
Then all the zeros of P(z) lie in z
1
[
an
n
L M
n
] .
II. MAIN RESULTS In this paper we prove the following generalization of Theorem B: 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, 0 n 1
and for some k 1 , k 2 1 , k1 k2
n 1
n 1
n
k1
n
n k2
n
2
n 1
...... k 1
1
2
n 1 ...... k 2
k 1
,
k2
,
and L 1 1 2 ...... 1 0 0 ,
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Location of Zeros of Polynomials M 1 1 2 ...... 1 0 0 .
Then all the zeros of P(z) lie in z
( k 1 1 )
i ( k 2 1) n
n
n
1
an
[
n ( k 1 1)
n
an
a0
R
z
X
1
log
log c
X a0
1
a0
log c 1
does not exceed
log
a0
j)
, c 1 , R 1 does not exceed a0
) and the number of zeros of P(z) in
R
z
Y
1
j
( k 2 1) n L M ].
n
a0
Y a0
log c
j ) ( k 2 1) (
j
c
X
log( 1
(
j 1
( k 1 1)
Further the number of zeros of P(z) in
Y
log( 1
log c
, c 1, R 1
c
) , where R is any positive number and
a0 n
X an R
n 1
R [( n
n
n ) (
) L M
0
0 ( k 1 1)
(
j
j)
j 1 n
( k 2 1)
(
j )] ,
j
j 1
n n 1
Y an R
R [(
n
n ) (
) L M
0
0 ( k 1 1)
(
j
j)
j 1 n
( k 2 1)
(
j )] .
j
j 1
k k
1
2 Remark 1: For 1 , Theorem 1 reduces to Theorem B. Taking k 2 1 in Theorem 1, we get the following result: n
Corollary
1:
P(z)
Let
j
a jz
be
a
polynomial
of
degree
with
n
j0
Re( a j ) j , Im( a j ) j , j 0 ,1 , 2 ,......,
n such that for some , , 0 n 1, 0 n 1
and for some k 1 1 , k1
n 1
n
k1
n
2
n 1
n n 1 ......
...... k 1
1
k 1
,
,
L 1 1 2 ...... 1 0 0 , M 1 1 2 ...... 1 0 0 .
Then all the zeros of P(z) lie in z ( k 1 1 )
n
1
an
n
[
n
n ( k 1 1)
(
j
j ) ( k 1 1)
L M
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n
j 1
].
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Location of Zeros of Polynomials a0
Also the number of zeros of P(z) in
R
z
X
1
log
X a0
log c
1
a0
X
log( 1
log c 1
does not exceed
log c
) and the number of zeros of P(z) in 1
a0
a0
z
R
Y
a0
Y a0
log
, c 1 , R 1 does not exceed
c
Y
log( 1
log c
, c 1, R 1
c
) , where R is any positive number and
a0 n
X an R
n 1
R [( n
n
n ) (
) L M
0
0 ( k 1 1)
(
j )] ,
j
j 1
n
Y an R
n 1
R [(
n ) (
n
) L M
0
0 ( k 1 1)
(
j )]
j
j 1
Taking 0 in Theorem 1, we get the following result: n
Corollary
2:
P(z)
Let
j
a jz
be
a
polynomial
of
degree
with
n
j0
Re( a j ) j , Im( a j ) j , j 0 ,1 , 2 ,......,
n such that for some , , 0 n 1, 0 n 1
and for some k 1 , k 2 1 , n
k1
n
k1
n
k2
n 1
k2
n
n 1
n 1
...... k 1 1 0 ,
n 1 ...... k 2 1 0 ,
Then all the zeros of P(z) lie in z
( k 1 1 )
i ( k 2 1) n
n
an
n
1
[
an
n
n ( k 1 1) (
a0
z
X
1
log
X a0
log c
1
a0
does not exceed
log( 1
log c 1
log
log c
j
j 1
( k 1 1)
Also the number of zeros of P(z) in
n
j ) ( k 2 1) (
R
, c 1 , R 1 does not exceed
c
X
) and the number of zeros of P(z) in
a0
z
Y
1
a0
j)
( k 2 1) n 0 0 0 0 ].
n
a0
Y a0
j
j 1
Y
log( 1
log c
R
, c 1, R 1
c
) , where R is any positive number and
a0 n
X an R
n 1
R [( n
n
n ) (
0
0) L M
0
0 ( k 1 1) (
j
j )] ,
j 1 n
( k 2 1) (
j )] , R 1
j
j 1 n
Y an R
n 1
R [(
n
n ) (
0
0) L M
0
0 ( k 1 1) (
j
j )]
j 1 n
( k 2 1) (
j
j )] , R 1
j 1
Similarly for other values of the parameters in Theorem 1, we get many other interesting results.
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Location of Zeros of Polynomials III. LEMMAS For the proof of Theorem 2, we make use of the following lemmas: Lemma 1: Let f(z) (not identically zero) be analytic for z R , f ( 0 ) 0 and f ( a k ) 0 , k 1, 2 ,......, n . Then
1 2
n
2
log f (Re
i
d log f ( 0 )
0
R
log
a
j 1
.
j
Lemma 1 is the famous Jensen’s Theorem (see page 208 of [1]). Lemma 2: Let f(z) be analytic , f ( 0 ) 0 and f ( z ) M for z R . Then the number of zeros of f(z) in z
R
1
, c 1 is less than or equal to
M
log
log c
c
.
f (0 )
Lemma 2 is a simple deduction from Lemma 1.
IV. PROOF OF THEOREM 1 Consider the polynomial F ( z ) (1 z ) P ( z )
(1 z )( a n z an z
n 1
n
a n 1 z
n 1
...... a 1 z a 0 )
( a n a n 1 ) z
...... ( a 1 a ) z
n
1
( a a 1 ) z
...... ( a 1 a 0 ) z a 0
anz
n 1
( k 1 1 ) n z
( k 1
1
( k 1 1 )(
i {( k 2 (k 2
( k 1
n
)z
z
n 1
1
n 1
n
n 1
1
)z
(
n2
n 1
n2
z
)z
( k 1
n
)z
1
( k 2 1 )(
n
n 1
n 1
1
(k 2 z
n2
)z
n 1
......
...... ( 1 0 ) z
.......
( k 2 1) n z
n
1
)z
n
n 1
z
1
)
n 1
0
......
n2
)z
1
z
n 1
1
......
) (
1
)z
...... ( 1 0 ) z 0 }
1
For z 1 so that
z
j
1, j 1, 2 ,......, n , we have, by using the hypothesis
F ( z ) a n z ( k 1 1 )
k 1
1
n
z
1
n 1
z
...... k 2
1
( k 2 1 )(
n 1
z
n
z
1
[ k 1
z
1
......
n 1
n
z [ a n z ( k 1 1 )
n 1
( k 1 1 )(
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n
i ( k 2 1) n z
......
1
z
1
i ( k 2 1)
n
z
n
z
n 1
n
k 1
...... 1 1
) k2 z
1 1
n
0
n 1
z
n 1
z
n
n2
z
n 1
......
0
k2
n 1
z
n2
n 1
...... 1 0 z 0
)]
[ k 1
n
n 1
k 1
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n 1
z
n2
k 1
n2
z
n3
2
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Location of Zeros of Polynomials k 1
......
z
1 n 1
) k2
z
n
) k2
1
n 1
n
k2
n
k 1
n 1
......
n
1
1
1
n
z
n 1
k 1
n2
( k 1 1 )(
0
n2
n3
......
n 1
1
)] n 1
n2
k 1
n2
...... k 1 1 1 ...... 1 0 0 ( k 1 1)(
......
z
n 1
)]
1
k 1
n 1
n 1
0
...... k 2
n2
n 1
1
( k 1 1 )(
n
1
n 1
z
...... 1
1
n 1
z
0
z
......
......
i ( k 2 1 ) n { k 1
n
z
n 1
...... 1 0 0 ( k 2 1 )(
z [ a n z ( k 1 1 )
n2
0
n 1
z
k2
n 1
1
......
z
k2
n
1
n
i ( k 2 1 ) n [ k 1
n
1
( k 2 1 )(
n
z
z [ a n z ( k 1 1 )
n 1
0
n 1
...... k 1
z
n
1 0
......
n 1
z
1
n3
.....
n 1
) k 2 n n 1 k 2 n 1 n 2 ...... k 2 1 1
...... 1 0 0 ( k 2 1 )( n 1 ...... 1 )}] n
z [ a n z ( k 1 1 ) ...... k 1 k2
n 1
1
i ( k 2 1 ) n { k 1
n
L ( k 1 1 )(
...... k 2
n2
1
n 1
n
k 1
n 1
......
1
n 1
) k2
M ( k 2 1 )(
n 1
n2
n
k 1
n2
n3
n 1
......
1
)}]
n n
z [ a n z ( k 1 1 )
i ( k 2 1)
n
{
n
n
n
( k 1 1)
(
j
j)
j 1 n
( k 2 1)
(
j ) ( k 1 1)
j
n
( k 2 1)
n
L M
}]
j 1
0
if n
a n z ( k 1 1 )
n
i ( k 2 1)
{
n
n
n
n
( k 1 1)
(
j ) ( k 2 1)
j
j 1
( k 1 1)
n
(
j
j)
j 1
( k 2 1) n L M }]
i.e.if z
( k 1 1 )
n
i ( k 2 1) n
1
an
n
[
an
n
n ( k 1 1)
n
(
j ) ( k 2 1)
j
j 1
( k 1 1)
n
(
j
j)
j 1
( k 2 1) n L M ].
This shows that those zeros of F(z) whose modulus is greater than 1 lie in z
( k 1 1 )
n
i ( k 2 1) n an
1 an
n
[
n
n ( k 1 1)
n
(
j
j ) ( k 2 1)
j 1
(
j
j)
j 1
( k 1 1) n ( k 2 1) n L M ].
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Location of Zeros of Polynomials 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 z
( k 1 1 )
n
i ( k 2 1) n
n
1
an
[
n
an
n
n ( k 1 1)
(
j ) ( k 2 1)
j
j 1
( k 1 1)
(
j)
j
j 1
( k 2 1) n L M ].
n
Again F (z) a0 G (z) ,
where G (z) an z
n 1
( k 1 1 ) n z
( k 1
1
( k 1 1 )(
i {( k 2 (k 2
n
1
( k 1
n
)z
n 1
z
n 1
n 1
n2
z
1
n2
( k 1
n
)z
)z
n 1
( k 2 1 )(
1
(k 2
n
z
n 1
)z
n2
n 1
......
...... ( 1 0 ) z
.......
( k 2 1) n z
n
1
)z
(
)z
n 1
1
n
n 1
1
z
n 1
)
......
n 1
)z
n2
z
1
1
......
) (
)z
1
...... ( 1 0 ) z }.
For z R , R 0 , we have G (z) an R
n 1
( k 1 1)
( k 1 1 ) R ( k 1 1)( (k 2
n
(k 2
1
n 1
)R
n
n 1
)R
1
1 R
n 1
1
n2
R
n2
( k 2 1)
n
)R
n 1
R
( k 1
n
R
n
( k 2 1 )(
n 1
n
(k 2
n 1
n2
)R
n 1
......
...... 1 0 R
....... 1 R
R
n
( k 1
n
R
n 1
n 1
1
n2
......
)
)R
n 1
1
R
1
...... )
1
R
...... 1 0 R }
an R
n 1
R [( k 1 1)
( k 1 1 )( k2
k 1 n n 1 k 1
n
n 1
n 1
n2
n
......
) k2
1
...... k 2
1
n
n 1
n 2 ...... k 1 1 L
n 1
M ( k 2 1 )(
n 1
......
1
)]
n
an R
n 1
R [( n
n
n ) (
) L M
0
0 ( k 1 1)
(
j
j)
j 1 n
( k 2 1)
(
j
j )]
j 1
X
for R 1 and for R 1 n
G (z) an R
n 1
R [(
n
n ) (
) L M
0
0 ( k 1 1)
(
j
j)
j 1
n
( k 2 1)
(
j
j )]
j 1
Y
Since G(0)=0 and G(z) is analytic for z R , it follows, by Schwarz Lemma , that for z R ,
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Location of Zeros of Polynomials G ( z ) X z for R 1 and G ( z ) Y z for R 1 .
Hence, for z R , R 1 F (z) a0 G (z) a0 G (z) a0 X z 0
if a0
z
.
X
Similarly, for z R , R 1 , F ( z ) 0 if z
a0
.
Y
In other words, F(z) does not vanish in z
a0
for R 1 and F(z) does not vanish in z
X
a0
for R 1
Y
in z R . That means all the zeros of F(z) and hence all the zeros of P(z) lie in z
a0
for R 1 and in
X z
a0
for R 1 in z R .
Y
Since for z R , F ( z ) X a 0 for R 1 and F ( z ) Y a 0 for R 1 and since F ( 0 ) a 0 0 , it follows by Lemma 2 that the number of zeros of P(z) in
not
1
exceed
log
log c a0 Y
z
R
X a0 a0
1 log c
, c 1 , R 1 does not exceed
c
log( 1
X
) and
the
a0
z
X
number
R
, c 1 , R 1 does
c
of
zeros
of
P(z)
in
a0
1
log
log c
Y a0 a0
1 log c
log( 1
Y
).
a0
That proves Theorem 1 completely.
REFERENCES [1]. [2]. [3]. [4]. [5]. [6]. [7]. [8]. [9]. [10]. [11].
L. V. Ahlfors, Complex Analysis, 3rd edition, Mc-Grawhill. A. Aziz and Q. G. Mohammad, Zero-free regions for polynomials and some generalizations of Enestrom-Kakeya Theorem, Canad. Math. Bull.,27(1984),265- 272. A. Aziz and B. A. Zargar, Some Extensions of Enestrom-Kakeya Theorem, Glasnik Mathematicki, 51 (1996), 239-244. Y.Choo, On the zeros of a family of self-reciprocal polynomials, Int. J. Math. Analysis , 5 (2011),1761-1766. N. K. Govil and Q. I. Rahman, On Enestrom-Kakeya Theorem, Tohoku J. Math. 20 (1968), 126-136. M. H. Gulzar, Some Refinements of Enestrom-Kakeya Theorem, International Journal of Mathematical Archive , Vol 2(9), 2011, 1512-1519.. M. H. Gulzar,B.A. Zargar and A. W. Manzoor, Zeros of a Certain Class of Polynomials, International Journal of Advanced Research in Engineering and Management, Vol.3 No. 2(2017). A. Joyal, G. Labelle and Q. I. Rahman, On the location of the zeros of a polynomial, Canad. Math. Bull. ,10 (1967), 53-63. M. Marden, Geometry of Polynomials, Math. Surveys No. 3, Amer. Math.Soc.(1966). Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, New York (2002). B. A. Zargar, On the zeros of a family of polynomials, Int. J. of Math. Sci. &Engg. Appls. , Vol. 8 No. 1(January 2014), 233-237.
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