Online ISSN: 2515-8260

Keywords : Polynomial


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

Barchand Chanam; Khangembam Babina Devi; Reingachan N; Thangjam Birkramjit Singh; Kshetrimayum Krishnadas

European Journal of Molecular & Clinical Medicine, 2020, Volume 7, Issue 2, Pages 1448-1455

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
 
 
 
 
1
1
max max
1
n
z R n z
R k
p z n p z
k

 

 

.
The result is best possible and extremal polynomial is    n
p z  z  k .
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.