- 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
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
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.
If we take in the previous Theorem, we get a result, which was proved by Aziz and Dawood .
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 .
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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