INTEGRAL INEQUALITY FOR HIGHER DERIVATIVE VERSIONS ON THEOREMS OF S. BERNSTEIN

. Let p ( z ) = n (cid:80) ν =0 a ν z ν be a polynomial of degree n and p (cid:48) ( z ) its derivative. If max | z | = r | p ( z ) | is denoted by M ( p, r ). If p ( z ) has all its zeros on | z | = k , k ≤ 1, then it was shown by Govil [3] that In this paper, we first prove a result concerning the s th derivative where 1 ≤ s < n of the polynomial involving some of the co-efficients of the polynomial. Our result not only improves and generalizes the above inequality, but also gives a generalization to higher derivative of a result due to Dewan and Mir [2] in this direction. Further, a direct generalization of the above inequality for the s th derivative where 1 ≤ s < n is also proved.