Lp INEQUALITIES FOR THE POLAR DERIVATIVE
OF A POLYNOMIAL
Lp Inequalities for the
Polar Derivative
NISAR A. RATHER
P.G. Department of Mathematics
Kashmir University
Hazratbal, Srinagar-190006, India
EMail: dr.narather@gmail.com
Nisar A. Rather
vol. 9, iss. 4, art. 103, 2008
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Contents
Received:
04 May, 2007
Accepted:
30 July, 2008
Communicated by:
JJ
II
S.S. Dragomir
J
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2000 AMS Sub. Class.:
26D10, 41A17.
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Key words:
Lp inequalities, Polar derivatives, Polynomials.
Abstract:
Let Dα P (z) denote the polar derivative of a polynomial P (z) of degree n with
respect to real or complex number α. If P (z) does not vanish in |z| < k, k ≥ 1,
then it has been proved that for |α| ≥ 1 and p > 0,
!
|α| + k
kDα P kp ≤
kP kp .
kk + zkp
An analogous result for the class of polynomials having no zero in |z| > k, k ≤ 1
is also obtained.
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Contents
1
Introduction and Statement of Results
3
2
Lemmas
10
3
Proofs of the Theorems
14
Lp Inequalities for the
Polar Derivative
Nisar A. Rather
vol. 9, iss. 4, art. 103, 2008
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1.
Introduction and Statement of Results
Let Pn (z) denote the space of all complex polynomials P (z) of degree n. For P ∈
Pn , define
Z 2π
p1
1
p
P (eiθ )
kP kp :=
,
1 ≤ p < ∞,
2π 0
and
kP k∞ := max |P (z)| .
|z|=1
Nisar A. Rather
vol. 9, iss. 4, art. 103, 2008
If P ∈ Pn , then
(1.1)
Lp Inequalities for the
Polar Derivative
kP 0 k∞ ≤ n kP k∞
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and
(1.2)
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0
kP kp ≤ nkP kp .
Inequality (1.1) is a well-known result of S. Bernstein (see [12] or [15]), whereas
inequality (1.2) is due to Zygmund [16]. Arestov [1] proved that the inequality
(1.2) remains true for 0 < p < 1 as well. Equality in (1.1) and (1.2) holds for
P (z) = az n , a 6= 0. If we let p → ∞ in (1.2), we get inequality (1.1).
If we restrict ourselves to the class of polynomials P ∈ Pn having no zero in
|z| < 1, then both the inequalities (1.1) and (1.2) can be improved. In fact, if P ∈ Pn
and P (z) 6= 0 for |z| < 1, then (1.1) and (1.2) can be, respectively, replaced by
n
(1.3)
kP 0 k∞ ≤ kP k∞
2
and
n
(1.4)
kP 0 kp ≤
kP kp ,
p ≥ 1.
k1 + zkp
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Inequality (1.3) was conjectured by P. Erdös and later verified by P. D. Lax
[10] whereas the inequality (1.4) was discovered by De Bruijn [5]. Rahman and
Schmeisser [13] proved that the inequality (1.4) remains true for 0 < p < 1 as well.
Both the estimates are sharp and equality in (1.3) and (1.4) holds for P (z) = az n +b,
|a| = |b| .
Malik [11] generalized inequality (1.3) by proving that if P ∈ Pn and P (z) does
not vanish in |z| < k where k ≥ 1, then
(1.5)
kP 0 k∞ ≤
n
kP k∞ .
1+k
Govil and Rahman [8] extended inequality (1.5) to the Lp -norm by proving that
if P ∈ Pn and P (z) 6= 0 for |z| < k where k ≥ 1, then
(1.6)
kP 0 kp ≤
n
kP kp ,
kk + zkp
Lp Inequalities for the
Polar Derivative
Nisar A. Rather
vol. 9, iss. 4, art. 103, 2008
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p ≥ 1.
It was shown by Gardner and Weems [7] and independently by Rather [14] that
the inequality (1.6) remains true for 0 < p < 1 as well.
Let Dα P (z) denote the polar derivative of polynomial P (z) of degree n with
respect to a real or complex number α. Then
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Dα P (z) = nP (z) + (α − z)P 0 (z).
Polynomial Dα P (z) is of degree at most n − 1. Furthermore, the polar derivative
Dα P (z) generalizes the ordinary derivative P 0 (z) in the sense that
Dα P (z)
= P 0 (z)
α→∞
α
lim
uniformly with respect to z for |z| ≤ R, R > 0.
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A. Aziz [2] extended inequalities (1.1) and (1.3) to the polar derivative of a polynomial and proved that if P ∈ Pn , then for every complex number α with |α| ≥ 1,
(1.7)
kDα P k∞ ≤ n |α| kP k∞
and if P ∈ Pn and P (z) 6= 0 for |z| < 1, then for every complex number α with
|α| ≥ 1,
(1.8)
kDα P k∞ ≤
n
(|α| + 1) kP k∞ .
2
Both the inequalities (1.7) and (1.8) are sharp. If we divide both sides of (1.7) and
(1.8) by |α| and let |α| → ∞, we get inequalities (1.1) and (1.3) respectively.
A. Aziz [2] also considered the class of polynomials P ∈ Pn having no zero in
|z| < k and proved that if P ∈ Pn and P (z) 6= 0 for |z| < k where k ≥ 1, then for
every complex number α with |α| ≥ 1,
|α| + k
kP k∞ .
(1.9)
kDα P k∞ ≤ n
1+k
The result is best possible and equality in (1.9) holds for P (z) = (z + k)n where α
is any real number with α ≥ 1.
It is natural to seek an Lp - norm analog of the inequality (1.7). In view of the Lp
- norm extension (1.2) of inequality (1.1), one would expect that if P ∈ Pn , then
(1.10)
kDα P kp ≤ n |α| kP kp ,
is the Lp - norm extension of (1.7) analogous to (1.2). Unfortunately, inequality
(1.10) is not, in general, true for every complex number α. To see this, we take in
Lp Inequalities for the
Polar Derivative
Nisar A. Rather
vol. 9, iss. 4, art. 103, 2008
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particular p = 2, P (z) = (1 − iz)n and α = iδ where δ is any positive real number
such that
p
n + 2n(2n − 1)
(1.11)
1≤δ<
,
3n − 2
then from (1.10), by using Parseval’s identity, we get, after simplication
n(1 + δ)2 ≤ 2(2n − 1)δ 2 .
This inequality can be written as
!
!
p
p
n + 2n(2n − 1)
n − 2n(2n − 1)
(1.12)
δ−
δ−
≥ 0.
3n − 2
3n − 2
!
p
2n(2n − 1)
1−
3n − 2
!
p
2(n − 1) + 2n(2n − 1)
>0
3n − 2
n−
=
and hence from (1.12), it follows that
!
p
n + 2n(2n − 1)
δ−
≥ 0.
3n − 2
This gives
δ≥
Nisar A. Rather
vol. 9, iss. 4, art. 103, 2008
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Since δ ≥ 1, we have
!
p
n − 2n(2n − 1)
δ−
≥
3n − 2
n+
Lp Inequalities for the
Polar Derivative
p
2n(2n − 1)
,
3n − 2
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which clearly contradicts (1.11). Hence inequality (1.10) is not, in general, true for
all polynomials of degree n ≥ 1.
While seeking the desired extension of inequality (1.8) to the Lp -norm, recently
Govil et al. [9] have made an incomplete attempt by claiming to have proved that if
P ∈ Pn and P (z) does not vanish in |z| < 1, then for every complex number α with
|α| ≥ 1, and p ≥ 1,
!
|α| + 1
(1.13)
kDα P kp ≤ n
kP kp .
k1 + zkp
A. Aziz, N.A. Rather and Q. Aliya [4] pointed out an error in the proof of inequality (1.13) given by Govil et al. [9] and proved a more general result which
not only validated inequality (1.13) but also extended inequality (1.6) for the polar
derivative of a polynomial P ∈ Pn . In fact, they proved that if P ∈ Pn and P (z) 6= 0
for |z| < k where k ≥ 1, then for every complex number α with |α| ≥ 1 and p ≥ 1,
!
|α| + k
(1.14)
kDα P kp ≤ n
kP kp .
kk + zkp
The main aim of this paper is to obtain certain Lp inequalities for the polar derivative of a polynomial valid for 0 < p < ∞. We begin by proving the following
extension of inequality (1.2) to the polar derivatives.
Theorem 1.1. If P ∈ Pn , then for every complex number α and p > 0,
(1.15)
kDα P kp ≤ n(|α| + 1) kP kp .
Remark 1. If we divide the two sides of (1.15) by |α| and make |α| → ∞, we get
inequality (1.2) for each p > 0.
Lp Inequalities for the
Polar Derivative
Nisar A. Rather
vol. 9, iss. 4, art. 103, 2008
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As an extension of inequality (1.6) to the polar derivative of a polynomial, we
next present the following result which includes inequalities (1.13) and (1.14) for
each p > 0 as a special cases.
Theorem 1.2. If P ∈ Pn and P (z) does not vanish in |z| < k where k ≥ 1, then for
every complex number α with |α| ≥ 1 and p > 0,
!
|α| + k
(1.16)
kDα P kp ≤ n
kP kp .
kk + zkp
Lp Inequalities for the
Polar Derivative
Nisar A. Rather
vol. 9, iss. 4, art. 103, 2008
In the limiting case, when p → ∞, the above inequality is sharp and equality in
(1.16) holds for P (z) = (z + k)n where α is any real number with α ≥ 1.
The following result immediately follows from Theorem 1.2 by taking k = 1.
Corollary 1.3. If P ∈ Pn and P (z) does not vanish in |z| < 1, then for every
complex number α with |α| ≥ 1 and p > 0,
!
|α| + 1
kP kp .
(1.17)
kDα P kp ≤ n
k1 + zkp
Remark 2. Corollary 1.3 not only validates inequality (1.13) for p ≥ 1 but also
extends it for 0 < p < 1 as well.
Remark 3. If we let p → ∞ in (1.16), we get inequality (1.9). Moreover, inequality
(1.6) also follows from Theorem 1.2 by dividing the two sides of inequality (1.16)
by |α| and then letting |α| → ∞.
We also prove:
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Theorem 1.4. If P ∈ Pn and P (z) has all its zeros in |z| ≤ k where k ≤ 1 and
P (0) 6= 0, then for every complex number α with |α| ≤ 1 and p > 0,
!
|α| + k
(1.18)
kDα P kp ≤ n
kP kp .
kk + zkp
In the limiting case, when p → ∞, the above inequality is sharp and equality in
(1.18) holds for P (z) = (z + k)n for any real α with 0 ≤ α ≤ 1.
The following result is an immediate consequence of Theorem 1.4.
Lp Inequalities for the
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Nisar A. Rather
vol. 9, iss. 4, art. 103, 2008
Corollary 1.5. If P ∈ Pn and P (z) has all its zeros in |z| ≤ k where k ≤ 1, then
for every complex number α with |α| ≤ 1,
|α| + k
kDα P k∞ ≤ n
kP k∞ .
1+k
The result is best possible and equality in (1.18) holds for P (z) = (z + k)n for any
real α with 0 ≤ α ≤ 1.
Finally, we prove the following result.
Theorem 1.6. If P ∈ Pn is self- inversive, then for every complex number α and
p > 0,
!
|α| + 1
kDα P kp ≤ n
kP kp .
k1 + zkp
The above inequality extends a result due to Dewan and Govil [6] for the polar
derivatives.
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2.
Lemmas
For the proof of these theorems, we need the following lemmas.
Lemma 2.1 ([2]). If P ∈ Pn and P (z) does not vanish in |z| < k where k ≥ 1, then
for every real or complex number γ with |γ| ≥ 1,
|Dγk P (z)| ≤ k Dγ/k Q(z) for |z| = 1
where Q(z) = z n P (1/z).
Lp Inequalities for the
Polar Derivative
Nisar A. Rather
Setting α = γk where k ≥ 1 in Lemma 2.1, we immediately get:
Lemma 2.2. If P ∈ Pn and P (z) does not vanish in |z| < k where k ≥ 1, then for
every real or complex number α with |α| ≥ 1,
|Dα P (z)| ≤ k Dα/k2 Q(z) for |z| = 1
where Q(z) = z n P (1/z) .
Lemma 2.3. If P ∈ Pn and P (z) 6= 0 in |z| < k where k ≥ 1 and Q(z) =
z n P (1/z), then for |z| = 1,
0
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0
k |P (z)| ≤ |Q (z)| .
Lemma 2.3 is due to Malik [9].
Lemma 2.4. If P ∈ Pn and P (z) 6= 0 in |z| < k where k ≥ 1 and Q(z) =
z n P (1/z), then for every real β, 0 ≤ β < 2π,
2 0
k P (z) + eiβ Q0 (z) ≤ k P 0 (z) + eiβ Q0 (z) for |z| = 1.
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Proof of Lemma 2.4. By hypothesis, P ∈ Pn and P (z) does not vanish in |z| < k
where k ≥ 1 and Q(z) = z n P (1/z). Therefore, by Lemma 2.3, we have
2
2
k 2 |P 0 (z)| ≤ |Q0 (z)|
for |z| = 1.
Multiplying both sides of this inequality by (k 2 − 1) and rearranging the terms, we
get
2
2
2
2
k 4 |P 0 (z)| + |Q0 (z)| ≤ k 2 |P 0 (z)| + k 2 |Q0 (z)| for |z| = 1.
Adding 2 Re k 2 P 0 (z)Q0 (z)eiβ to the both sides of (2.1), we obtain for |z| = 1,
2 0
k P (z) + eiβ Q0 (z)2 ≤ k 2 P 0 (z) + eiβ Q0 (z)2 for |z| = 1
(2.1)
Lp Inequalities for the
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Nisar A. Rather
vol. 9, iss. 4, art. 103, 2008
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and hence
2 0
k P (z) + eiβ Q0 (z) ≤ k P 0 (z) + eiβ Q0 (z) for |z| = 1.
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This proves Lemma 2.4.
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Lemma 2.5. If P ∈ Pn and Q(z) = z n P (1/z), then for every p > 0 and β real,
0 ≤ β < 2π,
Z 2π Z 2π
Z 2π
0 iθ
P (e ) + eiβ Q0 (eiθ )p dθdβ ≤ 2πnp
P (eiθ )p dθ.
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Lemma 2.6. If P ∈ Pn and P (z) does not vanish in |z| < k where k ≥ 1 and
Q(z) = z n P (1/z), then for every complex number α, β real, 0 ≤ β < 2π, and
p > 0,
Z 2π Z 2π
Z 2π
Dα P (eiθ ) + eiβ k 2 Dα/k2 Q(eiθ )p dθdβ ≤ 2πnp (|α| + k)p
P (eiθ )p dθ.
0
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Lemma 2.5 is due to the author [14] (see also [3]).
0
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Proof of Lemma 2.6. We have Q(z) = z n P (1/z), therefore, P (z) = z n Q(1/z) and
it can be easily verified that for 0 ≤ θ < 2π,
nP (eiθ ) − eiθ P 0 (eiθ ) = ei(n−1)θ Q0 (eiθ ) and nQ(eiθ ) − eiθ Q0 (eiθ ) = ei(n−1)θ P 0 (eiθ ).
Also, since P ∈ Pn and P (z) does not vanish in |z| < k, k ≥ 1, therefore, Q ∈ Pn .
Hence for every complex number α, β real, 0 ≤ β < 2π, we have
Dα P (eiθ ) + eiβ k 2 Dα/k2 Q(eiθ )
α
= (nP (eiθ ) + (α − eiθ )P 0 (eiθ ) + k 2 eiβ nQ(eiθ ) + 2 − eiθ Q0 (eiθ ) k
= (nP (eiθ ) − eiθ P 0 (eiθ ) + k 2 eiβ nQ(eiθ ) − eiθ Q0 (eiθ )
+ α P 0 (eiθ ) + eiβ Q0 (eiθ ) |
= ei(n−1)θ Q0 (eiθ ) + k 2 eiβ ei(n−1)θ P 0 (eiθ ) + α P 0 (eiθ ) + eiβ Q0 (eiθ ) ≤ |α| P 0 (eiθ ) + eiβ Q0 (eiθ ) + k 2 P 0 (eiθ ) + eiβ Q0 (eiθ ) .
This gives, with the help of Lemma 2.4,
Dα P (eiθ ) + eiβ k 2 Dα/k2 Q(eiθ ) ≤ |α| P 0 (eiθ ) + eiβ Q0 (eiθ )
+ k P 0 (eiθ ) + eiβ Q0 (eiθ )
= (|α| + k) P 0 (eiθ ) + eiβ Q0 (eiθ ) ,
which implies for each p > 0,
Z 2π Z 2π
Dα P (eiθ ) + eiβ k 2 Dα/k2 Q(eiθ )p dθdβ
0
0
Z 2π Z 2π
0 iθ
p
P (e ) + eiβ Q0 (eiθ )p dθdβ.
≤ (|α| + k)
0
0
Lp Inequalities for the
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Nisar A. Rather
vol. 9, iss. 4, art. 103, 2008
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Combining this with Lemma 2.5, we get
Z
0
2π
Z
2π
Dα P (eiθ ) + eiβ k 2 Dα/k2 Q(eiθ )p dθdβ
0
p
p
Z
≤ 2πn (|α| + k)
0
This completes the proof of Lemma 2.6.
2π
P (eiθ )p dθ.
Lp Inequalities for the
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vol. 9, iss. 4, art. 103, 2008
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3.
Proofs of the Theorems
Proof of Theorem 1.1. Let Q(z) = z n P (1/z), then P (z) = z n Q(1/z) and (as before) for 0 ≤ θ < 2π, we have
nP (eiθ )−eiθ P 0 (eiθ ) = ei(n−1)θ Q0 (eiθ ) and
nQ(eiθ )−eiθ Q0 (eiθ ) = ei(n−1)θ P 0 (eiθ ),
which implies for every complex number α and β real, 0 ≤ β < 2π,
Dα P (eiθ ) + eiβ nQ(eiθ ) + (α − eiθ )Q0 (eiθ ) = |nP (eiθ ) + (α − eiθ )P 0 (eiθ ) + eiβ nQ(eiθ ) − eiθ Q0 (eiθ ) + αQ0 (eiθ ) |
= | nP (eiθ ) − eiθ P 0 (eiθ ) + eiβ nQ(eiθ ) − eiθ Q0 (eiθ )
+ α P 0 (eiθ ) + eiβ Q0 (eiθ ) |
= |ei(n−1)θ Q0 (eiθ ) + eiβ ei(n−1)θ P 0 eiθ ) + α P 0 (eiθ ) + eiβ Q0 (eiθ ) |
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vol. 9, iss. 4, art. 103, 2008
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≤ |ei(n−1)θ Q0 (eiθ ) + eiβ ei(n−1)θ 0 (eiθ )| + |α| |P 0 (eiθ ) + eiβ Q0 (eiθ )|
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= (|α| + 1) |P 0 (eiθ ) + eiβ Q0 (eiθ )|.
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This gives with the help of Lemma 2.5 for each p > 0,
Z 2π Z 2π
Dα P (eiθ ) + eiβ nQ(eiθ ) + (α − eiθ )Q0 (eiθ ) p dθdβ
0
0
Z 2π Z 2π
0 iθ
p
P (e ) + eiβ Q0 (eiθ )p dθdβ
≤ (|α| + 1)
0
Z0 2π
P (eiθ )p dθ.
≤ 2πnp (|α| + 1)p
(3.1)
0
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Now using the fact that for any p > 0,
Z 2π
a + beiβ p dβ ≥ 2π max (|a|p , |b|p ) ,
0
(see [5, Inequality (2.1)]), it follows from (3.1) that
Z 2π
p1
Z 2π
p1
p
p
iθ
iθ
P (e ) dθ
Dα P (e ) dθ
≤ n (|α| + 1)
,
0
p > 0.
0
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Nisar A. Rather
This completes the proof of Theorem 1.1.
vol. 9, iss. 4, art. 103, 2008
Proof of Theorem 1.2. Since P ∈ Pn and P (z) does not vanish in |z| < k where
k ≥ 1, by Lemma 2.2, we have for every real or complex number α with |α| ≥ 1,
(3.2)
|Dα P (z)| ≤ k Dα/k2 Q(z)
for |z| = 1,
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n
where Q(z) = z P (1/z). Also, by Lemma 2.6, for every real or complex number
α, p > 0 and β real,
Z 2π Z 2π
iθ
iβ 2
iθ p
(3.3)
Dα P (e ) + e k Dα/k2 Q(e ) dβ dθ
0
0
Z 2π
p
p
P (eiθ )p dθ.
≤ 2πn (|α| + k)
0
0
R + eiβ p dβ ≥
Z
0
2π
r + eiβ p dβ,
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which implies
2π
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Now for every real β, 0 ≤ β < 2π and R ≥ r ≥ 1, we have
R + eiβ ≥ r + eiβ ,
Z
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p > 0.
If Dα P (eiθ ) 6= 0, we take R = k 2 Dα/k2 Q(eiθ ) / Dα P (eiθ ) and r = k, then by
(3.2), R ≥ r ≥ 1, and we get
Z 2π
|Dα P (eiθ ) + eiβ k 2 Dα/k2 Q(eiθ )|p dβ
0
p
Z p 2π k 2 Dα/k2 Q(eiθ ) iβ
iθ
dβ
= Dα P (e )
e
+
1
Dα P (eiθ )
0
p
Z 2π 2
k Dα/k2 Q(eiθ ) iβ
iθ p
= Dα P (e )
Dα P (eiθ ) e + 1 dβ
0
p
Z 2π 2
k Dα/k2 Q(eiθ ) p
iθ
iβ
+ e dβ
= Dα P (e )
Dα P (eiθ ) 0
Z 2π
p
k + eiβ p dβ.
≥ Dα P (eiθ )
0
For Dα P (eiθ ) = 0, this inequality is trivially true. Using this in (3.3), we conclude
that for every real or complex number α with |α| ≥ 1 and p > 0,
Z 2π
Z 2π
Z 2π
p
p
p
iβ
iθ
p
k + e dβ
Dα P (e ) dθ ≤ 2πn (|α| + k)
P (eiθ )p dθ,
0
0
0
which immediately leads to (1.16) and this completes the proof of Theorem 1.2.
Proof of Theorem 1.4. By hypothesis, all the zeros of polynomial P (z) of degree n
lie in |z| ≤ k where k ≤ 1 and P (0) 6= 0. Therefore, if Q(z) = z n P (1/z) , then
Q(z) is a polynomial of degree n which does not vanish in |z| < (1/k), where
(1/k) ≥ 1. Applying Theorem 1.2 to the polynomial Q(z), we get for every real or
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complex number β with |β| ≥ 1 and p > 0,
2π
Z
(3.4)
Dβ Q(eiθ )p dθ
p1
≤n
0
|β| + k1
z + 1 k p
! Z
2π
Q(eiθ )p dθ
p1
.
0
Now since
Q(eiθ ) = P (eiθ ) ,
and
z +
0 ≤ θ < 2π
Lp Inequalities for the
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Nisar A. Rather
1
= 1 kz + kk ,
p
k p k
vol. 9, iss. 4, art. 103, 2008
it follows that (3.4) is equivalent to
Z
(3.5)
0
2π
Dβ Q(eiθ )p dθ
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p1
≤n
k |β| + 1
kz + kkp
! Z
2π
P (eiθ )p dθ
p1
Contents
.
0
Also, we have for every β with |β| ≥ 1 and 0 ≤ θ < 2π,
Dβ Q(eiθ ) = nQ(eiθ ) + (β − eiθ )Q0 (eiθ )
inθ
iθ
i(n−1)θ
i(n−2)θ 0 iθ
iθ
iθ
= ne P (e ) + (β − e ) ne
P (e ) − e
P (e ) iθ
iθ
0
iθ
0
iθ
= β nP (e ) − e P (e ) + P (e ))
= β nP (eiθ ) − eiθ P 0 (eiθ ) + P 0 (eiθ )
= β D1/β P (eiθ ) .
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Using this in (3.5), we get for |β| ≥ 1,
Z
(3.6)
0
2π
p p1
iθ |β| D1/β P (e ) dθ
≤n
k |β| + 1
kz + kkp
! Z
2π
P (eiθ )p dθ
p1
,
p > 0.
0
Replacing 1/β by α so that |α| ≤ 1, we obtain from (3.6)
! Z
p1
Z 2π
p1
2π |α|
+
k
p
p
Dα P (eiθ ) dθ
P (eiθ ) dθ
,
≤n
kz
+
kk
0
0
p
Lp Inequalities for the
Polar Derivative
Nisar A. Rather
vol. 9, iss. 4, art. 103, 2008
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for |α| ≤ 1 and p > 0. This proves Theorem 1.4.
Proof of Theorem 1.6. Since P (z) is a self inversive polynomial of degree n, P (z) =
Q(z) for all z ∈ C where Q(z) = z n P (1/z). This gives for every complex number
α,
|Dα P (z)| = |Dα Q(z)| , z ∈ C
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so that
(3.7)
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Dα Q(eiθ )/Dα P (eiθ ) = 1,
0 ≤ θ < 2π.
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Also, since Q(z) is a polynomial of degree n, then
(3.8)
Dα Q(eiθ ) = nQ(eiθ ) − eiθ Q0 (eiθ ) + αQ0 (eiθ ).
Combining (3.1) and (3.8), it follows that for every complex number α and p > 0,
Z
2π
Z
(3.9)
0
2π
Dα P (eiθ ) + Dα Q(eiθ )p dθdβ
0
p
≤ 2πn (|α| + 1)
p
Z
2π
P (eiθ )p dθ.
0
Using (3.7) in (3.9) and proceeding similarly as in the proof of Theorem 1.2, we
immediately get the conclusion of Theorem 1.6.
Lp Inequalities for the
Polar Derivative
Nisar A. Rather
vol. 9, iss. 4, art. 103, 2008
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References
[1] V.V. ARESTOV, On integral inequalities for trigonometric polynomials and
their derivatives, Izv. Akad. Nauk SSSR Ser. Mat., 45 (1981), 3–22 [in Russian]
English translation: Math. USSR - Izv., 18 (1982), 1–17. Approx. Theory, 55
(1988), 183–193.
[2] A. AZIZ, Inequalities for the polar derivative of a polynomial, J. Approx. Theory, 55 (1988), 183–193.
Lp Inequalities for the
Polar Derivative
Nisar A. Rather
[3] A. AZIZ AND N.A. RATHER, Some Zygmund type Lq inequalities for polynomials, J. Math. Anal. Appl., 289 (2004), 14–29.
vol. 9, iss. 4, art. 103, 2008
[4] A. AZIZ, N.A. RATHER AND Q. ALIYA, Lq norm inequalities for the polar
derivative of a polynomial, Math. Ineq. and Appl., 11 (2008), 283–296.
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[5] N.G. De BRUIJN, Inequalities concerning polynomial in the complex domain,
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591–598.
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[6] K.K. DEWAN AND N.K. GOVIL, An inequality for self-inversive polynomials,
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[7] R.B. GARDNER AND A. WEEMS, A Bernstein-type L inequality for a certain class of polynomials, J. Math. Anal. Appl., 229 (1998), 472–478.
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[8] N.K. GOVIL AND Q.I. RAHMAN, Functions of exponential type not vanshing
in a half-plane and related polynomials , Trans. Amer. Math. Soc., 137 (1969),
501–517.
[9] N.K. GOVIL. G. NYUYDINKONG AND B. TAMERU, Some Lp inequalitites
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[10] P. D. LAX, Proof of a conjecture of P. Erdos on the derivative of a poplynomial,
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[11] M.A. MALIK, On the derivative of a polynomial, J. London Math. Soc., 1
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Lp Inequalities for the
Polar Derivative
[12] G.V. MILOVANOVIĆ. D.S. MITRINOVIĆ AND Th. RASSIAS, Topics in
Polynomials: Extremal Properties, Inequalities, Zeros, World Scientific, Singapore, 1994.
vol. 9, iss. 4, art. 103, 2008
Nisar A. Rather
Title Page
[13] Q.I. RAHMAN AND G. SCHMEISSER, Lp inequalitites for polynomial, J.
Approx. Theory, 53 (1998), 26–32.
[14] N.A. RATHER, Extremal properties and location of the zeros of polynomials,
Ph. D. Thesis, University of Kashmir, 1998.
[15] A.C. SCHAEFFER, Inequalities of A. Markoff and S.N. Bernstein for polynomials and related functions, Bull. Amer. Math. Soc., 47 (1941), 565–579.
[16] A. ZYGMUND, A remark on conjugate series, Proc. London Math. Soc., 34
(1932), 392–400.
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