ON THE ZEROS OF THE POLAR DERIVATIVE OF A POLYNOMIAL
*Gulzar, M. H.
Department of Mathematics, University
ARTICLE INFO ABSTRACT
In this paper we find bounds for the zeros of the polar derivative of a polynomial under certain conditions on the coefficients. Our results generalize many known results in this direction.
Copyright © 2016 Gulzar and Manzoor. This is an open access article distributed under the Creative Commons Att use, distribution, and reproduction in any medium, provided the original work is properly cited.
INTRODUCTION
For any polynomial P(z) of degree n, the polar derivative of P(z) )
( ) ( ) ( )
(z nP z z P z
P
D which is a polynomial of degree at most n
in the sense that lim D P(z)P(z)
. In the context of the famous Enestrom
all the zeros of the polynomial
n
j jz a z P
0
) (
find bounds for the zeros of DP(z)under certain conditions on its coefficients. In this connection recently proved the following results:
Theorem A: Let
n
j j jz a z P
0 )
( be a polynomial of degree n and
respect to a real number
such that1 2
1
0(n1)a (n2)a ...3an 2a
na
If
0
, then all the zeros of the polar derivative*Corresponding author: Gulzar, M. H.
Department of Mathematics, University of Kashmir, Srinagar, India.
ISSN: 0975-833X
Vol.
Article History:
Received 18th November, 2015 Received in revised form 14th December, 2015 Accepted 25th January, 2016 Published online 27th February,2016
Citation:Gulzar, M. H. and Manzoor, A. W. 2016
(02), 26669-26674.
Key words:
Bound, coefficients, Polynomial, Zeros.
Mathematics Subject Classification:
30C10, 30C15.
RESEARCH ARTICLE
ON THE ZEROS OF THE POLAR DERIVATIVE OF A POLYNOMIAL
Gulzar, M. H. and Manzoor, A. W.
Department of Mathematics, University of Kashmir, Srinagar, India
ABSTRACT
In this paper we find bounds for the zeros of the polar derivative of a polynomial under certain conditions on the coefficients. Our results generalize many known results in this direction.
is an open access article distributed under the Creative Commons Attribution License, which use, distribution, and reproduction in any medium, provided the original work is properly cited.
, the polar derivative of P(z) with respect to a positive real number
which is a polynomial of degree at most n-1. The polar derivative generalizes the ordinary derivative
. In the context of the famous Enestrom-Kakeya Theorem (
jwith
...
0
0 1
1
a
a
a
a
n n lie inz
1
under certain conditions on its coefficients. In this connection recently
be a polynomial of degree n and DP(z)nP(z)(z)P(z)be a polar derivative of P(z) with
1
2
n
n a
a .
, then all the zeros of the polar derivative D0P(z)lie in 1 [ 1 0 0]
1
na na a a
z n
n
.
of Mathematics, University of Kashmir, Srinagar, India.
Available online at http://www.journalcra.com
International Journal of Current Research
Vol. 8, Issue, 02, pp.26669-26674 February, 2016
INTERNATIONAL
2016. “On the Zeros of the Polar Derivative of a Polynomial”, International Journal of Current
ON THE ZEROS OF THE POLAR DERIVATIVE OF A POLYNOMIAL
India
In this paper we find bounds for the zeros of the polar derivative of a polynomial under certain conditions on the coefficients. Our results generalize many known results in this direction.
ribution License, which permits unrestricted
with respect to a positive real number
is defined by 1. The polar derivative generalizes the ordinary derivative(Marden, 1966) which states that
1
, attempts have been made tounder certain conditions on its coefficients. In this connection recently Ramulu et al., (2015)
be a polar derivative of P(z) with
INTERNATIONAL JOURNAL OF CURRENT RESEARCH
Theorem B: Let
n
j j jz a z P
0 )
( be a polynomial of degree n and DP(z)nP(z)(z)P(z)be a polar derivative of P(z) with
respect to a real number
such that1 2 1 2
1
0(n1)a (n2)a ...3an 2an an
na .
If
0, then all the zeros of the polar derivative D0P(z)lie in 1 [ ]1 0 0 1
n
n
a na na a
z .
Later Reddy et al. (2015) proved the following generalizations of Theorems A and B:
Theorem C: Let
n
j j jz a z P
0 )
( be a polynomial of degree n and DP(z)nP(z)(z)P(z)be a polar derivative of P(z) with
respect to a real number
such that2 ,..., 2 , 1 , 0 , ) ( ) 1 ( )} 1 ( { )
2
(i ai2 n i ai1 i ai1 ni ai i n . Then all the zeros of the polar derivative DP(z)lie in
]
)
(
[
1
0 1 0 1 1 1
na
a
na
a
a
a
n
a
a
n
z
n nn n
.Theorem D: Let
n
j j
j
z
a
z
P
0
)
(
be a polynomial of degree n andD
P
(
z
)
nP
(
z
)
(
z
)
P
(
z
)
be a polar derivative ofP(z) with respect to a real number
such that2
,...,
2
,
1
,
0
,
)
(
)
1
(
)}
1
(
{
)
2
(
i
a
i2
n
i
a
i1
i
a
i1
n
i
a
ii
n
.Then all the zeros of the polar derivative
D
P
(z
)
lie in]
)
(
[
1
1 0
1 0
1 1
n nn n
a
a
n
na
a
na
a
a
a
n
z
.In this paper we find lower bounds for the zeros of
D
P
(
z
)
under the same conditions. In fact, we prove the following results:Theorem 1: Let
n
j j
j
z
a
z
P
0
)
(
be a polynomial of degree n andD
P
(
z
)
nP
(
z
)
(
z
)
P
(
z
)
be a polar derivative ofP(z) with respect to a real number
such that2
,...,
2
,
1
,
0
,
)
(
)
1
(
)}
1
(
{
)
2
(
i
a
i2
n
i
a
i1
i
a
i1
n
i
a
ii
n
.Then all the zeros of the polar derivative
D
P
(z
)
lie inM
na
a
z
1
0 , where)
(
)
(
1 1 01
n
a
a
a
na
a
a
n
M
n
n
n
n
Combining Theorem C and Theorem 1, we get the following result:
Theorem 2: Let
n
j j
j
z
a
z
P
0
)
(
be a polynomial of degree n andD
P
(
z
)
nP
(
z
)
(
z
)
P
(
z
)
be a polar derivative ofP(z) with respect to a real number
such that2
,...,
2
,
1
,
0
,
)
(
)
1
(
)}
1
(
{
)
2
(
i
a
i2
n
i
a
i1
i
a
i1
n
i
a
ii
n
.Then all the zeros of the polar derivative
D
P
(z
)
lie in],
)
(
[
1
0 1 0
1 1 1
0 1
na
a
na
a
a
a
n
a
a
n
z
M
na
a
n n n
n
where M is as given in Theorem 1.
Corollary 1: Let
n
j j
j
z
a
z
P
0
)
(
bea
polynomial of degree n with positive coefficients and)
(
)
(
)
(
)
(
z
nP
z
z
P
z
P
D
be a polar derivative of P(z) with respect to a real number
such that2
,...,
2
,
1
,
0
,
)
(
)
1
(
)}
1
(
{
)
2
(
i
a
i2
n
i
i1
i
a
i1
n
i
a
ii
n
.Then all the zeros of the polar derivative
D
P
(z
)
lie in1
)
(
)
(
2
1 1 00 1
z
na
a
a
a
n
na
a
n
n
.
Theorem 3: Let
n
j j
j
z
a
z
P
0
)
(
be a polynomial of degree n andD
P
(
z
)
nP
(
z
)
(
z
)
P
(
z
)
be a polar derivative ofP(z) with respect to a real number
such that2
,...,
2
,
1
,
0
,
)
(
)
1
(
)}
1
(
{
)
2
(
i
a
i2
n
i
a
i1
i
a
i1
n
i
a
ii
n
.Then all the zeros of the polar derivative
D
P
(z
)
lie in 1 0,
M
na
a
z
where.
0 1 1
1
n
a
a
a
na
a
a
n
M
n
n
n
n
Combining Theorem D and Theorem 3, we get the following result:
Theorem 4: Let
n
j j
j
z
a
z
P
0
)
(
be a polynomial of degree n andD
P
(
z
)
nP
(
z
)
(
z
)
P
(
z
)
be a polar derivative ofP(z) with respect to a real number
such that2
,...,
2
,
1
,
0
,
)
(
)
1
(
)}
1
(
{
)
2
(
i
a
i2
n
i
a
i1
i
a
i1
n
i
a
ii
n
.Then all the zeros of the polar derivative
D
P
(z
)
lie in]
)
(
[
1
1 0
1 0
1 1 0
1
n n n
n
a
a
n
na
a
na
a
a
a
n
z
M
na
a
,
where
M
is as given in Theorem 3Taking
,
a
i
0
,
i
0
,
1
,
2
,...,
n
in Theorem 3, we get the following result:Corollary 2: Let
n
j j
j
z
a
z
P
0
)
(
be a polynomial of degree n with positive coefficients and)
(
)
(
)
(
)
(
z
nP
z
z
P
z
P
D
be a polar derivative of P(z) with respect to a real number
such that2
,...,
2
,
1
,
0
,
)
(
)
1
(
)}
1
(
{
)
2
(
0
i
a
i2
n
i
i1
i
a
i1
n
i
a
ii
n
.Then all the zeros of the polar derivative
D
P
(z
)
lie in]
)
(
2
[
1
1
1 0 11
n nn n
a
a
n
na
a
a
a
n
z
.Proofs of Theorems
Proof of Theorem 1: We have
0 1 2 2 1
1
...
)
(
z
a
z
a
z
a
z
a
z
a
P
n nn
n
)
(
)
(
)
(
)
(
z
nP
z
z
P
z
P
D
3 3 2
2 2 1
1
1
)
{(
1
)
2
}
{(
2
)
3
}
(
n n nn n n
n n
n
a
z
n
a
a
z
n
a
a
z
a
n
)
(
}
)
1
(
2
{
}
)
2
(
3
{
...
a
3
n
a
2z
2
a
2
n
a
1z
a
1
na
0
.Now , consider the polynomial
)
(
)
1
(
)
(
z
z
D
P
z
1 1
2 1
1
)
{
(
1
)
2
}
{(
1
)
(
n n n n n n nn
a
z
n
a
n
a
a
z
n
a
a
n
)
(
}
)
1
(
2
{
}
)
1
(
)
2
2
(
3
{
...
}
3
)
2
2
(
0 1 0 1 2 2 1 2 3 2 3 2na
a
z
na
a
n
a
z
a
n
a
n
a
z
a
a
n
n n n
)
(
)
(
a
1
na
0
G
z
, where 1 1 2 11
)
{
(
1
)
2
}
{(
1
)
(
)
(
n n n n n n nn
a
z
n
a
n
a
a
z
n
a
a
n
z
G
z
na
a
n
a
z
a
n
a
n
a
z
a
a
n
n n n}
)
1
(
2
{
}
)
1
(
)
2
2
(
3
{
...
}
3
)
2
2
(
0 1 2 2 1 2 3 2 3 2
For
z
1
, we have by using the hypothesis,
(
1
)
2
(
1
)
(
2
2
)
(
z
n
a
n
a
n1
n
a
n
n
a
n1
a
n2
n
a
n1
G
0 1 2 1 2 3 3 2)
1
(
2
)
1
(
)
2
2
(
3
...
3
)
na
a
n
a
a
n
a
n
a
a
a
n
n n
2 11
(
1
)
2
n
a
na
nn
a
n
n
a
na
n0 1 2 1 2 3 3 2 1
)
1
(
2
)
1
(
)
2
2
(
3
...
3
)
2
2
(
)
1
(
na
a
n
a
a
n
a
n
a
a
a
n
a
n
n n n
)
(
)
(
1 1 01
n
a
a
a
na
a
a
n
n
n
n
n
M
Since G(z) is analytic for
z
1
and G(0)=0, it follows by Schwarz Lemma thatG
(
z
)
M
z
forz
1
.Hence, for
z
1
,)
(
)
(
)
(
z
a
1na
0G
z
F
0
)
(
0 1 0 1
z
M
na
a
z
G
na
a
ifM
na
a
z
1
0 .This shows that all the zeros of F(z) lie in
M
na
a
z
1
0 .Since the zeros of
D
P
(z
)
are also the zeros of F(z), it follows that all the zeros ofD
P
(z
)
lie inM
na
a
z
1
0 and theproof of Theorem 1 is complete.
Proof of Theorem 3: We have
0 1 2 2 1 1
...
)
(
z
a
z
a
z
a
z
a
z
a
P
n n
n n
)
(
)
(
)
(
)
(
z
nP
z
z
P
z
P
D
3 3 2 2 2 1 1
1
)
{(
1
)
2
}
{(
2
)
3
}
(
)
(
}
)
1
(
2
{
}
)
2
(
3
{
...
2 1 1 02 2
3
n
a
z
a
n
a
z
a
na
a
.Now, consider the polynomial
)
(
)
1
(
)
(
z
z
D
P
z
F
1 1
2 1
1
)
{
(
1
)
2
}
{(
1
)
(
n nn n
n n
n
n
a
z
n
a
n
a
a
z
n
a
a
n
)
(
}
)
1
(
2
{
}
)
1
(
)
2
2
(
3
{
...
}
3
)
2
2
(
0 1 0 1 2 2 1 2 3 2 3 2na
a
z
na
a
n
a
z
a
n
a
n
a
z
a
a
n
n n n
)
(
)
(
a
1
na
0
G
z
, where 1 1 2 11
)
{
(
1
)
2
}
{(
1
)
(
)
(
n n n n n n nn
a
z
n
a
n
a
a
z
n
a
a
n
z
G
z
na
a
n
a
z
a
n
a
n
a
z
a
a
n
n n n}
)
1
(
2
{
}
)
1
(
)
2
2
(
3
{
...
}
3
)
2
2
(
0 1 2 2 1 2 3 2 3 2
For
z
1
, we have by using the hypothesis,
(
1
)
2
(
1
)
(
2
2
)
(
z
n
a
n
a
n1
n
a
n
n
a
n1
a
n2
n
a
n1
G
0 1 2 1 2 3 3 2)
1
(
2
)
1
(
)
2
2
(
3
...
3
)
na
a
n
a
a
n
a
n
a
a
a
n
n n
1 3 1 21
2
(
1
)
3
(
1
)
n
a
na
na
nn
a
n
n
a
na
nn
a
n1 2 0 2 3 1 2
)
1
(
2
)
2
2
(
3
)
1
(
...
)
2
2
(
a
n
a
na
a
n
a
a
n
a
n
n
.
0 1 1 1M
na
a
a
a
n
a
a
n
n n n n
Since G(z) is analytic for
z
1
and G(0)=0, it follows by Schwarz Lemma thatG
(
z
)
M
z
forz
1
.Hence, for
z
1
,)
(
)
(
)
(
z
a
1na
0G
z
F
0
)
(
0 1 0 1
z
M
na
a
z
G
na
a
ifM
na
a
z
1 0 .This shows that all the zeros of F(z) lie in
M
na
a
z
1 0 .Since the zeros of
D
P
(z
)
are also the zeros of F(z), it follows that all the zeros ofD
P
(z
)
lie inM
na
a
z
1 0 and theREFERENCES
Marden, M., Geometry of Polynomials, Math. Survey No.3, Amer. Math. Society, RI Providence, 1966.
Ramulu, P. and G. L. Reddy, On the Zeros of Polar Derivatives, International Journal of Recent Research in Mathematics, Computer Science and Information Technology, Vol.2, Issue 1 (April 2015- September 2015), 143-145.
Reddy, G. L., P. Ramulu and C. Gangadhar, 2015. On the Zeros of Polar Derivatives of Polynomials, Journal of Research in Applied Mathematics (Quest Journals), Vol.2, Issue 4, 7-10.
