Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Inequalities for the Polar Derivative of a Polynomial | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 515709 9 pages doi 2009 515709 Research Article Inequalities for the Polar Derivative of a Polynomial M. Bidkham M. Shakeri and M. Eshaghi Gordji Department of Mathematics Faculty of Natural Sciences Semnan University Semnan 35195-363 Iran Correspondence should be addressed to M. Bidkham mdbidkham@ Received 11 August 2009 Accepted 30 November 2009 Recommended by Narendra Kumar Govil Let p z be a polynomial of degree n and for any real or complex number a and let Dap z np z a - z p z denote the polar derivative of the polynomial p z with respect to a. 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. Copyright 2009 M. Bidkham et al. 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. 1. Introduction and Statement of Results If p z is a polynomial of degree n then it is well known that maxip z n maxip z I _ I -1 11 I I I -1 11 z 1 z 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 p z has all its zeros at the origin. If p z 0 in z 1 then max p z nmax p z 1 1 1 x 9 II Í r X z 1 2 z 1 Inequality was conjectured by Erdos and later proved by Lax 1 . If the polynomial p z of degree n has all its zeros in z 1 then it was proved by Turan 2 that 2 Journal of Inequalities and Applications max p z nmaxip z z 1 2 z 1 Inequality was generalized by Malik 3 who proved that if p z 0 in z k k 1 then n max p z max p z z 1 1 k z i For the class of polynomials having all its zeros in z k k 1
