- Research Article
- Open Access
Inequalities for the Polar Derivative of a Polynomial
© M. Bidkham et al. 2009
- Received: 11 August 2009
- Accepted: 30 November 2009
- Published: 7 December 2009
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.
- General Class
- Previous Theorem
- Previous Inequality
- Polar Derivative
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 generalized by Malik  who proved that if in , then
For the class of polynomials having all its zeros in , Govil  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.  proved that
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  that if has all its zeros in , , then for ,
Inequality (1.9) was later sharpened by Dewan and Upadhye , who proved the following theorem.
Theorem 1 A.
Recently, Dewan et al.  extented inequality (1.6) to the polar derivative of a polynomial and obtained the following result.
Theorem 1 B.
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.
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.
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 .
Next we consider a class of polynomial having no zeros in , where and prove the following generalization of Theorem B.
For Theorem 1.3 reduces to Theorem B.
Dividing the two sides of (1.16) by and letting , we obtain a result of Chanam and Dewan .
For the proofs of these theorems we need the following lemmas.
This lemma is due to Aziz and Rather .
Inequality (2.2) is best possible and equality holds for .
This lemma is according to Aziz .
This lemma is according to Dewan et al. .
This result is according to Dewan et al. .
Lemma 2.5 is according to Chanam and Dewan .
Proof of Theorem 1.1.
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.
This completes the proof of the theorem.
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