**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 *D*_{}*P*(*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(*n*1)*a* (*n*2)*a* ...3*an* 2*a*

*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 (

*j*with

### ...

### 0

0 1

1

###

###

###

###

###

*a*

_{}

*a*

*a*

*a*

_{n}*lie in*

_{n}*z*

###

### 1

under certain conditions on its coefficients. In this connection recently

be a polynomial of degree n and *D**P*(*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 *D*0*P*(*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 *D**P*(*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(*n*1)*a* (*n*2)*a* ...3*an* 2*an* *an*

*na* .

If

###

0, then all the zeros of the polar derivative*D*

_{0}

*P*(

*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 *D**P*(*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* *ai*2 *n* *i* *ai*1 *i* *ai*1 *n**i* *ai* *i* *n* .
Then all the zeros of the polar derivative *D*_{}*P*(*z*)lie in

### ]

### )

### (

### [

### 1

0 1 0 1 1 1

*na*

*a*

*na*

*a*

*a*

*a*

*n*

*a*

*a*

*n*

*z*

_{n}

_{n}*n*
*n*

###

###

###

###

###

###

###

_{}

###

###

###

###

.**Theorem D:** Let

##

###

*n*

*j*
*j*

*j*

*z*

*a*

*z*

*P*

0

### )

### (

be a polynomial of degree n and*D*

_{}

*P*

### (

*z*

### )

###

*nP*

### (

*z*

### )

###

### (

###

###

*z*

### )

*P*

###

### (

*z*

### )

be a polar derivative ofP(z) with respect to a real number

###

such that### 2

### ,...,

### 2

### ,

### 1

### ,

### 0

### ,

### )

### (

### )

### 1

### (

### )}

### 1

### (

### {

### )

### 2

### (

*i*

###

###

*a*

_{i}_{}

_{2}

###

*n*

###

*i*

###

###

*a*

_{i}_{}

_{1}

###

*i*

###

###

*a*

_{i}_{}

_{1}

###

*n*

###

*i*

*a*

_{i}*i*

###

*n*

###

.Then all the zeros of the polar derivative

*D*

_{}

*P*

### (z

### )

lie in### ]

### )

### (

### [

### 1

1 0

1 0

1 1

###

###

###

###

###

###

###

*n*

*n*

*n*
*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 and*D*

_{}

*P*

### (

*z*

### )

###

*nP*

### (

*z*

### )

###

### (

###

###

*z*

### )

*P*

###

### (

*z*

### )

be a polar derivative ofP(z) with respect to a real number

###

such that### 2

### ,...,

### 2

### ,

### 1

### ,

### 0

### ,

### )

### (

### )

### 1

### (

### )}

### 1

### (

### {

### )

### 2

### (

*i*

###

###

*a*

_{i}_{}

_{2}

###

*n*

###

*i*

###

###

*a*

_{i}_{}

_{1}

###

*i*

###

###

*a*

_{i}_{}

_{1}

###

*n*

###

*i*

*a*

_{i}*i*

###

*n*

###

.Then all the zeros of the polar derivative

*D*

_{}

*P*

### (z

### )

lie in*M*

*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 and*D*

_{}

*P*

### (

*z*

### )

###

*nP*

### (

*z*

### )

###

### (

###

###

*z*

### )

*P*

###

### (

*z*

### )

be a polar derivative ofP(z) with respect to a real number

###

such that### 2

### ,...,

### 2

### ,

### 1

### ,

### 0

### ,

### )

### (

### )

### 1

### (

### )}

### 1

### (

### {

### )

### 2

### (

*i*

###

###

*a*

_{i}_{}

_{2}

###

*n*

###

*i*

###

###

*a*

_{i}_{}

_{1}

###

*i*

###

###

*a*

_{i}_{}

_{1}

###

*n*

###

*i*

*a*

_{i}*i*

###

*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

### )

### (

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 that### 2

### ,...,

### 2

### ,

### 1

### ,

### 0

### ,

### )

### (

### )

### 1

### (

### )}

### 1

### (

### {

### )

### 2

### (

*i*

###

###

*a*

_{i}_{}

_{2}

###

*n*

###

*i*

###

###

_{i}_{}

_{1}

###

*i*

###

###

*a*

_{i}_{}

_{1}

###

*n*

###

*i*

*a*

_{i}*i*

###

*n*

###

.Then all the zeros of the polar derivative

*D*

_{}

*P*

### (z

### )

lie in### 1

### )

### (

### )

### (

### 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 and*D*

*P*

### (

*z*

### )

###

*nP*

### (

*z*

### )

###

### (

###

###

*z*

### )

*P*

###

### (

*z*

### )

be a polar derivative ofP(z) with respect to a real number

###

such that### 2

### ,...,

### 2

### ,

### 1

### ,

### 0

### ,

### )

### (

### )

### 1

### (

### )}

### 1

### (

### {

### )

### 2

### (

*i*

###

###

*a*

_{i}_{}

_{2}

###

*n*

###

*i*

###

###

*a*

_{i}_{}

_{1}

###

*i*

###

###

*a*

_{i}_{}

_{1}

###

*n*

###

*i*

*a*

_{i}*i*

###

*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 and*D*

_{}

*P*

### (

*z*

### )

###

*nP*

### (

*z*

### )

###

### (

###

###

*z*

### )

*P*

###

### (

*z*

### )

be a polar derivative ofP(z) with respect to a real number

###

such that### 2

### ,...,

### 2

### ,

### 1

### ,

### 0

### ,

### )

### (

### )

### 1

### (

### )}

### 1

### (

### {

### )

### 2

### (

*i*

###

###

*a*

_{i}_{}

_{2}

###

*n*

###

*i*

###

###

*a*

_{i}_{}

_{1}

###

*i*

###

###

*a*

_{i}_{}

_{1}

###

*n*

###

*i*

*a*

_{i}*i*

###

*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 that### 2

### ,...,

### 2

### ,

### 1

### ,

### 0

### ,

### )

### (

### )

### 1

### (

### )}

### 1

### (

### {

### )

### 2

### (

### 0

###

*i*

###

###

*a*

_{i}_{}

_{2}

###

*n*

###

*i*

###

###

_{i}_{}

_{1}

###

*i*

###

###

*a*

_{i}_{}

_{1}

###

*n*

###

*i*

*a*

_{i}*i*

###

*n*

###

.Then all the zeros of the polar derivative

*D*

_{}

*P*

### (z

### )

lie in### ]

### )

### (

### 2

### [

### 1

### 1

_{1}

_{0}

_{1}

1

###

###

###

###

###

###

_{n}

_{n}*n*
*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*

*n*

*n*

*n*

###

###

###

###

###

###

### )

### (

### )

### (

### )

### (

### )

### (

*z*

*nP*

*z*

*z*

*P*

*z*

*P*

*D*

_{}

###

###

###

###

###

3 3 2

2 2 1

1

1

### )

### {(

### 1

### )

### 2

### }

### {(

### 2

### )

### 3

### }

### (

###

###

###

###

###

###

###

###

*n*

*n*

*n*

*n*
*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*

_{2}

*z*

2 ###

*a*

_{2}

###

*n*

###

*a*

_{1}

*z*

###

*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*

*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 2*na*

*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*

*n*

*n*

*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*

*n*1

###

*n*

*a*

*n*

###

###

###

*n*

*a*

*n*1

###

*a*

*n*2

###

*n*

###

*a*

*n*1

###

###

*G*

0
1
2
1
2
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1

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and G(0)=0, it follows by Schwarz Lemma that*G*

### (

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*M*

*z*

for *z*

###

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.Hence, for

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1###

0**.**

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*M*

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1###

0**.**

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*D*

_{}

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### (z

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are also the zeros of F(z), it follows that all the zeros of*D*

_{}

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lie in*M*

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1###

0 and theproof of Theorem 1 is complete.

**Proof of Theorem 3: **We have

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###

###

Since G(z) is analytic for

*z*

###

### 1

and G(0)=0, it follows by Schwarz Lemma that*G*

### (

*z*

### )

###

*M*

*z*

for *z*

###

### 1

.Hence, for

*z*

###

### 1

,### )

### (

### )

### (

### )

### (

*z*

*a*

1 *na*

0 *G*

*z*

*F*

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### (

0 1 0 1###

###

###

###

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###

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*G*

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*a*

###

###

if*M*

*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 of*D*

_{}

*P*

### (z

### )

lie in*M*

*na*

*a*

*z*

###

###

###

###

1 0 and the**REFERENCES **

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.