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 | |
/
| |
| |
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
| |
/
| |
| |
(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
| |
| |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
| |
| |
If we divide both sides of inequality (1.8) by | | and letting | | ∞ and noting that / We get the following ∞
| |
| |
| |
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
| |
| |
| |
1
| |
| |
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,
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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)