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International Journal of Scientific and Research Publications, Volume 3, Issue 2, February 2013 764
ISSN 2250-3153
Theorem 1 : Let
1≤
k ≤ n,
≠ 0,
If
a
and
If
and
a n −k
ρ
n− k −1 > an−k
δ
1
a n
a
P(
z)
=
n
∑
j=
0
a j
z
j
be a polynomial of degree n such that for some real numbers
+ an ≥ an− 1
≥ ... ≥ an−k
+ 1
≥ λan−k
≥ an−k
−1
≥ ... ≥ a1
≥ µ a0
.
,
then P(z) has all its zeros in the disk
K
( λ −1)
γ
1
=
a
k + 1
an
− k
− δ K
1
k
− γ
1
= 0
n ,
2 ρ +
n
+ ( λ −1)
an− k
− µ ( a0
+ a0
) + 2 a
=
a
,
− k
>
n−k
+1
a
0
n
then P(z) has all its zeros in the disk
K
(1 − λ)
γ
2
=
a
k
− δ
an
− k
γ
k −1
2
K −
2
=
0
z ≤ K 1
, where
K1
is the greatest positive root of the equation
. .
n ,
2 ρ + an
+ (1 − λ)
an− k
− µ ( a0
+ a0
) + 2 a0
δ
2
=
an
.
Remark 1: Taking ρ = 0 , µ = 1,
Theorem 1 reduces to Theorem B.
Taking
ρ = 0
, Theorem 1 gives the following result:
Corollory 1: Let
1≤
k ≤ n,
≠ 0,
If
a
and
If
and
a n −k
a
n− k −1 > an−k
δ =
1
a n
a
≥
P(
z)
=
n
∑
j=
0
a j
z
j
ρ ≥ 0,0
< µ ≤1
, λ ≠ 1, ,
z ≤ K 2
, where
K
2 is the greatest positive root of the equation
be a polynomial of degree n such that for some real number
1
≥ ... ≥ ≥ ≥ ≥ ... ≥ ≥ a
n
an− an−k
+ 1
λan−k
an−k
−1
a1
µ
0
.
,
then P(z) has all its zeros in the disk
K
( λ −1)
γ
1
=
a
k + 1
an
− k
− δ K
1
k
− γ
1
= 0
n ,
+ ( λ −1)
a − ( a + a ) + 2 a
an
n− k
µ
0 0
0
,
− k
>
n−k
+1
a
n
then P(z) has all its zeros in the disk
K
(1 − λ)
γ
2
=
a
n
k
an
− k
− δ K
,
γ
k −1
2
−
2
=
0
z ≤ K 1
, where
K1
is the greatest positive root of the equation
. .
z ≤ K 2
, where
K
2 is the greatest positive root of the equation
0 < µ ≤ 1
, ≠ 1
λ , ,
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