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Inequalities for the Polar Derivative of a Polynomial
Journal of Inequalities and Applications volume 2009, Article number: 515709 (2009)
Abstract
Let be a polynomial of degree
and for any real or complex number
, and let
denote the polar derivative of the polynomial
with respect to
. In this paper, we obtain new results concerning the maximum modulus of a polar derivative of a polynomial with restricted zeros. Our results generalize as well as improve upon some well-known polynomial inequalities.
1. Introduction and Statement of Results
If is a polynomial of degree
, then it is well known that

The above inequality, which is an immediate consequence of Bernstein's inequality applied to the derivative of a trigonometric polynomial, is best possible with equality holding if and only if has all its zeros at the origin. If
in
, then

Inequality (1.2) was conjectured by Erdös and later proved by Lax [1]. If the polynomial of degree
has all its zeros in
, then it was proved by Turán [2] that

Inequality (1.2) was generalized by Malik [3] who proved that if in
, then

For the class of polynomials having all its zeros in , Govil [4] proved that

Inequality (1.5) is sharp and equality holds for . By considering a more general class of polynomials
, not vanishing in
,
, Gardner et al. [5] proved that

where and
Let denote the polar derivative of the polynomial
of degree
with respect to the point
. Then

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

As an extension of (1.5), it was shown by Aziz and Rather [6] that if has all its zeros in
,
, then for
,

Inequality (1.9) was later sharpened by Dewan and Upadhye [7], who proved the following theorem.
Theorem 1 A.
Let be a polynomial of degree
having all its zeros in
,
, then for
,

where
Recently, Dewan et al. [8] extented inequality (1.6) to the polar derivative of a polynomial and obtained the following result.
Theorem 1 B.
If is a polynomial of degree
having no zeros in
, then for
,

where and
In this paper, we will first generalize Theorem A as well as improve upon the bound obtained in inequality (1.10) by involving some of the coefficients of . More precisely, we prove the following.
Theorem 1.1.
If is a polynomial of degree
having all its zeros in
, then for
,

for and

for , where
.
Now it is easy to verify that if , then
,
and
for
. Hence for polynomial of degree
, Theorem 1.1 is a refinement of Theorem A.
Dividing both sides of inequalities (1.12) and (1.13) by and letting
, we get the following result.
Corollary 1.2.
If is a polynomial of degree
having all its zeros in
,
, then

for and

for , where
.
These inequalities are sharp and equality holds for the polynomial .
If we take in the previous Theorem, we get a result, which was proved by Aziz and Dawood [9].
Next we consider a class of polynomial having no zeros in , where
and prove the following generalization of Theorem B.
Theorem 1.3.
If is a polynomial of degree
having no zeros in
, then for
and
,

where

Remark 1.4.
For Theorem 1.3 reduces to Theorem B.
Remark 1.5.
Dividing the two sides of (1.16) by and letting
, we obtain a result of Chanam and Dewan [10].
2. Lemmas
For the proofs of these theorems we need the following lemmas.
Lemma 2.1.
If has all its zeros in
, then for every
,

where .
This lemma is due to Aziz and Rather [6].
Lemma 2.2.
If is a polynomial of degree n, having all its zeros in
, where
, then

Inequality (2.2) is best possible and equality holds for .
This lemma is according to Aziz [11].
Lemma 2.3.
If is a polynomial of degree
, then for
,

if , and

if .
This lemma is according to Dewan et al. [12].
Lemma 2.4.
If is a polynomial of degree
having no zeros in
and
, then for
,

if , and

if .
This result is according to Dewan et al. [13].
Lemma 2.5.
If is a polynomial of degree
such that
in
,
, then for
,

where
Lemma 2.5 is according to Chanam and Dewan [10].
3. Proof of the Theorems
Proof of Theorem 1.1.
By hypothesis that the polynomial has all its zeros in
, where
, therefore all the zeros of the polynomial
lie in
. Applying Lemma 2.1 to the polynomial
and noting that
, we get

that is,

The polynomial is of degree
and so
is the polynomial of degree
, where
, hence applying Lemma 2.3 to the polynomial
, we get for

Combining (3.2) and (3.3), we get for

Since the polynomial has all zeros in
the polynomial
has no zero in
, hence the polynomial
has all its zeros in
, therefore on applying Lemma 2.4 to the polynomial
, we get

Since (and similarly for the minima), (3.5) is equivalent to

Combining (3.4) and (3.6) we get the desired result. This completes the proof of inequality (1.12). The proof of the Theorem in the case follows along the same lines as the proof of (1.12) but instead of inequalities (2.3) and (2.5), we use inequalities (2.4) and (2.6), respectively.
Proof of Theorem 1.3.
By hypothesis that the polynomial has no zero in
, where
, therefore the polynomial
has no zero in
, where
. Since
, using Theorem B we have

where and

Using Lemma 2.5 in the previous inequality, we get

This completes the proof of the theorem.
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Bidkham, M., Shakeri, M. & Eshaghi Gordji, M. Inequalities for the Polar Derivative of a Polynomial. J Inequal Appl 2009, 515709 (2009). https://doi.org/10.1155/2009/515709
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DOI: https://doi.org/10.1155/2009/515709