HIGHER DERIVATIVE VERSIONS ON THEOREMS OF S. BERNSTEIN

Let $p(z)=\sum\limits_{\nu=0}^{n}a_{\nu}z^{\nu}$ be a polynomial of degree n and $p^{\prime}(z)$ its derivative. If $\max\limits_{{\mid}z{\mid}=r}{\mid}p(z){\mid}$ 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 $$M(p^{\prime},\;1){\leq}\frac{n}{k^n+k^{n-1}}M(p,\;1)$$. In this paper, we first prove a result concerning the sth 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 sth derivative where 1 ≤ s < n is also proved.