Cmjv02i01p0139

J. Comp. & Math. Sci. Vol.2 (1), 139-143 (2011)

Inequalities for the Polar Derivative of A Polynomial M. S. PUKHTA Division of Agri. Engineering S. K. University of Agricultural Sciences and Technology of Kashmir, Srinagar-191121, India email: mspukhta_67@yahoo.co.in ABSTRACT Let be a polynomial of degree having all its zeros in | | 1 . It was proved by Aziz and Dawood1 that

| |

/

| |

| |

| |

| |

For the polynomial having all its zeros in | | 1 with s-fold zeros at origin, Shah2 proved that

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/

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In this paper we shall extend both the inequalities to the polar derivative and there by present a compact generalization of these results.

Key words and phrases: Polynomials, Polar Derivative, inequalities, Zeros.

1. INTRODUCTION AND STATEMENT OF RESULTS Let be a polynomial of degree having all its zeros in the unit disk | | 1 , then it was shown by Turan6 that

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/

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(1.1)

Inequality (1.1) is best possible with

where | | | | is the extremal polynomial.

As a refinement of (1.1) Aziz and Dawood1 proved that if has all its zeros in | | 1, then

/ | |

| | | |

| |

| |

(1.2)

The equality in (1.2) holds for | | | | . Shah2 where

generalized (1.1) and proved that if has all its zeros in | | 1 with S-fold zeros at the origin , then

| |

/

| |

| |

(1.3)

Inequality (1.3) is best possible and equality holds for the polynomial

where 0 .

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

M. S. Pukhta, J. Comp. & Math. Sci. Vol.2 (1), 139-143 (2011)

140

Let denote the polar differentiation of the polynomial of degree with respect to the point . Then

/

The polynomial is of degree at most 1 and it generalizes the ordinary derivative in the sense that

∞ / (1.4)

Shah4 extended (1.1) to the polar derivative of and proved that if all the zeros of the polynomial lie in | | 1 , then

| | | |

| | 1 | |

| | , | | 1 (1.5)

Aziz and Rather3 generalized (1.5) and proved that if all the zeros of lie in | | , 1 , then for every real or complex number with | |

| |

| | | |

ŕł™ ŕ´‹

ŕ´‹

1 .

| |

Theorem 1.2. If is a polynomial of | |

| |

| |

(1.6)

In the same paper Aziz and Rather3 sharpened the inequality (1.5) by proving that if all the zeros of lie in | | 1 then for every real or complex number

with | | 1

| |

| |

| | 1

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| |1

| |

| |

| |

(1.7)

In this paper we shall extend (1.5) and (1.3) to the polar derivative and thereby present a compact generalization of these results.

Firstly we prove the following Theorem 1.1. If is a polynomial of degree having all its zeros in | | , 1 , then for every real or complex number with | | , | | ŕł™

ŕł™ ŕł™ ŕ´‹

| |

ŕł™

| | , | |

and Dawood1.

Corollary 1.1. If is a polynomial of degree having all its zeros in | | , 1 , then for every real or complex number with | | ,

1 1

1 | | | | ' ! "

# $ % 1&

| | | |

1 1

Next we prove a result which extends (1.3) to the polar derivative.

| |

| |

(1.8)

corollary, which is an improvement of Aziz

/ | | 1 2

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| |

If we divide both sides of inequality (1.8) by | | and letting | | ∞ and noting that

/ We get the following ∞

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| |

| |

degree having all its zeros in | | 1 with s-fold zeros at the origin then for every real or complex number with | | , we have

| |

| |

| |

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

(1.9)

M. S. Pukhta, J. Comp. & Math. Sci. Vol.2 (1), 139-143 (2011)

where 1 and 0 .

If we divide both sides (1.9) by | | and | | ∞ letting and noting that

/

, then our result ∞

reduces to Shah2.

For the proof of Theorem1 and 2 we need the following lemmas.

Lemma 2.1. If ( ∑ Âľ , 1 , is a polynomial of degree and * | | , , 1 and 0 for if

| | then for - . 1 | |

| | | |

/ 0

inequality (1.7) to the polynomial 4 and

noting that

/ 0

| |

| | (2.1)

The above lemma is due to Dewan,Naresh Singh and Abdullah Mir5.

Lemma 2.2. Let ( ∑ Âľ , 1 , is a polynomial of degree having all its zeros in | | 1 , 1 1 , then

/ 2 / | | 1 , 333333

where 2 / 0 .

| |

| | for

(2.2)

3. PROOF OF THE THEOREMS Proof of Theorem 1.1.

Since has all its zeros in | | , where 1 , therefore, all the zeros of 4 lie in | | 1. Applying

1,

| |

We get

| | | |

2. LEMMAS

141

ŕ´€ ŕł–

1

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| |

| |

1

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| |

Which implies

/ | |

| | | |

| | | | | |

| |

Which gives | |

| | / 0 | | | |

| | / 0 | | | |5

| | (3.1)

Again since p z has all its zeros in |z| k, k 1 , therefore, the polynomial 3333

q z z p / 0 has all its zeros in |z|

and hence the polynomial q / 0 has all its zeros in |z| 1. Applying Lemma 1 to the

polynomial q / 0 for k 1, we get

Âľ

!"# | | :q / 0: / Âľ 0

!$ / Âľ 0 | | :q / 0:

!"# | |

Which is equivalent to Âľ

!"# |p z |

/

Âľ 0 | |

/ Âľ 0

!$ | |

|p z |

:q / 0:

!"# | |

|p z |

!"# | |

|p z |

Which implies !"# |p z | | |

!$ / Âľ 0 | |

k /

Âľ Âľ0

|p z |

(3.2) Also if F z is a polynomial of degree , then (see[7,pp 337-347])

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

M. S. Pukhta, J. Comp. & Math. Sci. Vol.2 (1), 139-143 (2011)

142 !"# | | %&

|F z | R

!"# | |

|F z |

Applying this result to the polynomial D' p z np z Îą z p/ z , which is of degree n 1 , it follows that for k 1, max max |D p z | k |D p z | |z| $ k |z| $ 1

(3.3)

Using (3.2) and (3.3) in (3.1) , we get

| | | |

| |

ŕ´‹

| | ŕł™

| | $ 1 and | |

ŕ´‹

ŕł™

| |

| | | |

Which proves the desired result.

Proof of Theorem 1.2. Since has all its zeros in | | 1, with s-fold zeros at the origin, we write @ where @ is a polynomial of degree having all its zeros in | | 1. It can be easily 3333 3333

verified that if 2 / 0 @ / 0 then

( / $ / for | | $ 1

(3.4)

Now has all its zeros in | | 1 , therefore, by Lemma 2 we have

/

( / ೙డഋ

| |

| | )*+ | | $ 1

೙డഋ | |

| | for

(3.7)

Now / @ @/ which implies

+ / , 0 ,

ŕł™ ŕ´‹ | | | |

ŕł™

ŕł™ ŕ´‹

| | | | /

-E /

| |

| |

Combining (3.4),(3.5) and (3.6) we get

If , , ) , ‌ ‌ ‌ ‌ , are the zeros of @ then * 1 , 1 B , we have for 0 C , 2D

Which implies

| |

(3.6)

/ -/ $,

-

| | | | ŕł™ ŕ´‹ ŕł™ | | | |

| | ŕł™ ŕ´‹ | |

| | $ /

| | /

/

Also for every real or complex number

with | | , we have

+ -/ , 0 - , + = ∑

* -E !+ "

∑* .

.

For point E , , 0 C , 2D , other than zeros of . Hence we have / FE , G

FE , G

(3.8)

For points E , , 0 C , 2D , other than zeros of . Since inequality (3.8) is true for the points E , , 0 C , 2D , which are the zeros of also, it follows that /

(3.5)

-E /

| | HIJ | | 1

(3.9)

Inequality (3.9) in conjunction with (3.7) gives | | | |

| | | | | |

Which proves the result.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

M. S. Pukhta, J. Comp. & Math. Sci. Vol.2 (1), 139-143 (2011)

REFERENCES 1. A.Aziz & Q.M.Dawood, Inequalities for a polynomial and its derivative, Jour.approx. Theory, Vol. 54 , PP. 306313 (1988).

2. W. M. Shah, Extremal properties & bounds for the zeros of polynomials, Ph.D thesis, University of Kashmir, (1998).

3. A. Aziz & N. A. Rather , A refinement of a theorem of Paul Turan concerning polynomials , J. Math. Inequality,

143

Vol. 1, PP. 231-238 (1998).

4. W. M. Shah, A generalization of a theorem of Paul Turan, J. Ramanugan Math. Soc. Vol. 1, PP. 67-72 (1996).

5. K. K. Dewan, Naresh Singh & Abdullah Mir, Int. Jour. of Math. Analysis, Vol 1, No 11, PP 529-538 (2007).

6. P.Turan, Uber die ableitung von polynomen , Composition Math, Vol. 7, PP. 89-95 (1939).

7. M. Reisz, Uber einen satz de Herrn Serge Bernstein, Acta Math, 40, 337-347 (1918).

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)