On the Zeros of the Polar Derivative of a Polynomial

Full text

(1)

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 that

1 2

1

0(n1)a (n2)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 in

z

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 to

under 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

(2)

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 that

1 2 1 2

1

0(n1)a (n2)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 that

2 ,..., 2 , 1 , 0 , ) ( ) 1 ( )} 1 ( { )

2

(i ai2 ni ai1 i ai1 ni ai in . 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 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 of

P(z) with respect to a real number

such that

2

,...,

2

,

1

,

0

,

)

(

)

1

(

)}

1

(

{

)

2

(

i

a

i2

n

i

a

i1

i

a

i1

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 of

P(z) with respect to a real number

such that

2

,...,

2

,

1

,

0

,

)

(

)

1

(

)}

1

(

{

)

2

(

i

a

i2

n

i

a

i1

i

a

i1

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 0

1

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 of

P(z) with respect to a real number

such that

2

,...,

2

,

1

,

0

,

)

(

)

1

(

)}

1

(

{

)

2

(

i

a

i2

n

i

a

i1

i

a

i1

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.

(3)

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

i2

n

i

i1

i

a

i1

n

i

a

i

i

n

.

Then all the zeros of the polar derivative

D

P

(z

)

lie in

1

)

(

)

(

2

1 1 0

0 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 of

P(z) with respect to a real number

such that

2

,...,

2

,

1

,

0

,

)

(

)

1

(

)}

1

(

{

)

2

(

i

a

i2

n

i

a

i1

i

a

i1

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 of

P(z) with respect to a real number

such that

2

,...,

2

,

1

,

0

,

)

(

)

1

(

)}

1

(

{

)

2

(

i

a

i2

n

i

a

i1

i

a

i1

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 3

Taking

,

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

i2

n

i

i1

i

a

i1

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

(4)

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 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

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 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 1

1

(

1

)

2

n

a

n

a

n

n

a

n

n

a

n

a

n

0 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 0

1

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 that

G

(

z

)

M

z

for

z

1

.

Hence, for

z

1

,

)

(

)

(

)

(

z

a

1

na

0

G

z

F

0

)

(

0 1 0 1

z

M

na

a

z

G

na

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

proof 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

}

(

(5)

)

(

}

)

1

(

2

{

}

)

2

(

3

{

...

2 1 1 0

2 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 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 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

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 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 2

1

2

(

1

)

3

(

1

)

n

a

n

a

n

a

n

n

a

n

n

a

n

a

n

n

a

n

1 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 1

M

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 that

G

(

z

)

M

z

for

z

1

.

Hence, for

z

1

,

)

(

)

(

)

(

z

a

1

na

0

G

z

F

0

)

(

0 1 0 1

z

M

na

a

z

G

na

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

(6)

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.

Figure

Updating...

References

Updating...

Download now (6 pages)