Online ISSN: 2515-8260

Improved and Generalized Bernstein Type Inequalities for the Higher Derivatives of a Polynomial

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Barchand Chanam* , Khangembam Babina Devi, Reingachan N, Thangjam Birkramjit SinghKshetrimayum Krishnadas


Abstract- Let p(z) be a polynomial of degree n having no zero zero in z  k , k  1 , then for 1  R  k , Dewan and Bidkham [J. Math. Anal. Appl., 166(1992), 319-324] proved max z R pz R  k n1  n 1 k  n max z 1 p  z  . The result is best possible and extremal polynomial is pz  z  k  n . In this paper, by involving certain coefficients of the polynomial p(z) , we prove a result concerning the estimate of maximum modulus of higher derivatives of p(z) , which not only improves as well as generalizes the above result, but also has interesting consequences as special cases.

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