European Journal of Molecular & Clinical Medicine2515-82607220201123Improved and Generalized Bernstein Type Inequalities for the Higher Derivatives of a Polynomial144814552351ENBarchand ChanamDepartment of Mathematics
National Institute of Technology Manipur, Imphal, Manipur, IndiaKhangembam Babina DeviDepartment of Mathematics
National Institute of Technology Manipur, Imphal, Manipur, IndiaReingachan NDepartment of Mathematics
National Institute of Technology Manipur, Imphal, Manipur, IndiaThangjam Birkramjit SinghDepartment of Mathematics
National Institute of Technology Manipur, Imphal, Manipur, IndiaKshetrimayum KrishnadasDepartment of Mathematics
Shaheed Bhagat Singh College (University of Delhi),Delhi, IndiaJournal Article20201123Let p(z) be a polynomial of degree n having no zero zero in z k , k 1, then<br />for 1 R k , Dewan and Bidkham [J. Math. Anal. Appl., 166(1992), 319-324] proved<br /> <br /> <br /> <br /> <br />1<br />1<br />max max<br />1<br />n<br />z R n z<br />R k<br />p z n p z<br />k<br /><br /> <br /><br /> <br /><br />.<br />The result is best possible and extremal polynomial is n<br />p z z k .<br />In this paper, by involving certain coefficients of the polynomial p(z) , we prove a result<br />concerning the estimate of maximum modulus of higher derivatives of p(z) ,<br />which not only<br />improves as well as generalizes the above result, but also has interesting consequences as<br />special cases.https://ejmcm.com/article_2351_4a96b3c45fb505932baed07f4405dc66.pdf