Inequalities for the Polar Derivative of a Polynomial

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

(1.1)

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

(1.2)

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

(1.3)

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

(1.4)

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

(1.5)

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

(1.6)

where and

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

(1.7)

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

(1.8)

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

(1.9)

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 ,

(1.10)

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 ,

(1.11)

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 ,

(1.12)

for and

(1.13)

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

(1.14)

for and

(1.15)

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 ,

(1.16)

where

(1.17)

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 ,

(2.1)

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

(2.2)

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 ,

(2.3)

if , and

(2.4)

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 ,

(2.5)

if , and

(2.6)

if .

This result is according to Dewan et al. [13].

Lemma 2.5.

If is a polynomial of degree such that in , , then for ,

(2.7)

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

(3.1)

that is,

(3.2)

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

(3.3)

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

(3.4)

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

(3.5)

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

(3.6)

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

(3.7)

where and

(3.8)

Using Lemma 2.5 in the previous inequality, we get

(3.9)

This completes the proof of the theorem.