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.
1. Introduction and Statement of Results
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 .
3. Proof of the Theorems
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.
- Lax PD: Proof of a conjecture of P. Erdös on the derivative of a polynomial. Bulletin of the American Mathematical Society 1944, 50: 509–513. 10.1090/S0002-9904-1944-08177-9MathSciNetView ArticleMATHGoogle Scholar
- Turán P: Über die Ableitung von polynomen. Compositio Mathematica 1939, 7: 89–95.MathSciNetGoogle Scholar
- Malik MA: On the derivative of a polynomial. Journal of the London Mathematical Society 1969, 1: 57–60. 10.1112/jlms/s2-1.1.57View ArticleMATHGoogle Scholar
- Govil NK: On the derivative of a polynomial. Proceedings of the American Mathematical Society 1973, 41: 543–546. 10.1090/S0002-9939-1973-0325932-8MathSciNetView ArticleMATHGoogle Scholar
- Gardner RB, Govil NK, Weems A: Some results concerning rate of growth of polynomials. East Journal on Approximations 2004,10(3):301–312.MathSciNetMATHGoogle Scholar
- Aziz A, Rather NA: A refinement of a theorem of Paul Turán concerning polynomials. Mathematical Inequalities & Applications 1998,1(2):231–238.MathSciNetView ArticleMATHGoogle Scholar
- Dewan KK, Upadhye CM: Inequalities for the polar derivative of a polynomial. Journal of Inequalities in Pure and Applied Mathematics 2008,9(4, article 119):-9.Google Scholar
- Dewan KK, Singh N, Mir A: Extensions of some polynomial inequalities to the polar derivative. Journal of Mathematical Analysis and Applications 2009,352(2):807–815. 10.1016/j.jmaa.2008.10.056MathSciNetView ArticleMATHGoogle Scholar
- Aziz A, Dawood QM: Inequalities for a polynomial and its derivative. Journal of Approximation Theory 1988,54(3):306–313. 10.1016/0021-9045(88)90006-8MathSciNetView ArticleMATHGoogle Scholar
- Chanam B, Dewan KK: Inequalities for a polynomial and its derivative. Journal of Mathematical Analysis and Applications 2007,336(1):171–179. 10.1016/j.jmaa.2007.02.029MathSciNetView ArticleMATHGoogle Scholar
- Aziz A: Inequalities for the derivative of a polynomial. Proceedings of the American Mathematical Society 1983,89(2):259–266. 10.1090/S0002-9939-1983-0712634-5MathSciNetView ArticleMATHGoogle Scholar
- Dewan KK, Kaur J, Mir A: Inequalities for the derivative of a polynomial. Journal of Mathematical Analysis and Applications 2002,269(2):489–499. 10.1016/S0022-247X(02)00030-6MathSciNetView ArticleMATHGoogle Scholar
- Dewan KK, Singh N, Mir A: Growth of polynomials not vanishing inside a circle. International Journal of Mathematical Analysis 2007,1(9–12):529–538.MathSciNetMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.