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Báo cáo hóa học: " On the maximum modulus of a polynomial and its polar derivative"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : On the maximum modulus of a polynomial and its polar derivative | Zireh Journal of Inequalities and Applications 2011 2011 111 http www.journalofinequalitiesandapplications.eom content 2011 1 111 3 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access On the maximum modulus of a polynomial and its polar derivative Ahmad Zireh Correspondence azireh@shahroodut.ac.ir Department of Mathematics Shahrood University of Technology Shahrood Iran Abstract For a polynomial p z of degree n having all zeros in z 1 Jain is shown that max D. .D D. p z fr- q - 1 X 1 011- 1 z 1 2 1 at - 1 - 1 at - 1 min p z ail 1 2 1 K 1 t n . In this paper the above inequality is extended for the polynomials having all zeros in z k where k 1. Our result generalizes certain well-known polynomial inequalities. 2010 Mathematics Subject Classification. Primary 30A10 Secondary 30C10 30D15. Keywords Polar derivative Polynomial Inequality Maximum modulus Zeros 1. Introduction and statement of results Let p z be a polynomial of degree n then according to the well-known Bernstein s inequality 1 on the derivative of a polynomial we have max p z 1 n max p z . 11 Z 1 1 1 Z 1 1 1 This result is best possible and equality holding for a polynomial that has all zeros at the origin. If we restrict to the class of polynomials which have all zeros in z 1 then it has been proved by Turan 2 that max h z n max p z . 1.2 z 1 1 1 2 z 1 11 The inequality 1.2 is sharp and equality holds for a polynomial that has all zeros on z 1 As an extension to 1.2 Malik 3 proved that if p z has all zeros in z k where k 1 then Springer 2011 Zireh licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http creativecommons.org licenses by 2.0 which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Zireh Journal of Inequalities and Applications 2011 2011 111 http www.journalofinequalitiesandapplications.eom content 2011 1 111 Page 2 of 9 max p