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.

aν z ν be a polynomial of degree n and p (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 M (p , 1) ≤ 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.

Introduction
Let p(z) = n ν=0 a ν z ν be a polynomial of degree n in the complex plane and p (z) its derivative.
The result is best possible and equality in (1.1) holds for polynomials p(z) = αz n with |α| = 1.
If we restrict ourselves to the class of polynomials having no zero inside the unit disc |z| = 1, then Erdös conjectured and later Lax [5] proved Inequality (1.2) is best possible and the extremal polynomial is p(z) = α+βz n , where |α| = |β|.
a ν z ν is a polynomial of degree n not vanishing in |z| < k, k ≥ 1, then In the literature, there is no inequality analogous to (1.3) in the case p(z) = 0 in |z| < k, k ≤ 1. While trying to obtain an analogous inequality to (1.3) for this class of polynomials, Govil [3] was, in particular, only able to prove the following.

Lemmas
The following lemmas are needed for the proofs of the theorem and the corollary.
a ν z ν is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then for 0 ≤ θ < 2π and 1 ≤ s < n, where, q(z) = z n p 1 z . Proof. The polynomial P (z) = p(kz) has all its zeros in |z| ≤ 1 and hence the polynomial Q(z) = z n P 1 z = z n p k z = k n q z k has all its zeros in |z| ≥ 1.
also has all its zeros in |z| < 1, which implies that In particular, for |z| = k, since k ≥ 1,  (2.1) a ν z ν is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then for 1 ≤ s < n, Proof. By Lemma 2.1, Here p (s) (z) is a polynomial of degree at most (n − s), and applying Lemma 2.2 with R = k 2 ≥ 1, we have a ν z ν is a polynomial of degree n having no zero in |z| < k, k ≤ 1, then for 1 ≤ s < n, Proof. Since p(z) has no zero in |z| < k, k ≤ 1, q(z) has all its zeros in |z| ≤ 1 k , and hence the lemma follows readily.
where 1 ≤ s < n. Proof. The proof follows immediately on applying Lemma 2.7 to q(z) as in the proof of (2.8) of Lemma 2.6.

Inequality (3.3) when combined with Lemma 2.4 we obtain
which completes the proof of the theorem.

Remark 3.2.
Under the same hypothesis of Theorem 3.1, we claim that C(n, s)|a n |k 2s + |a n−s |k s−1 C(n, s)|a n |(k s−1 + k 2s ) + |a n−s |(1 + k s−1 ) where C(n, s) is as stated in Theorem 3.1. Which on simplification gives |a n−s | ≤ C(n, s)|a n |k s , and is true by inequality (2.9) of Lemma 2.6.
If we use inequality (3.4) to (3.1) of Theorem 3.1, we obtain the following direct generalization of inequality (1.4) to the sth derivative, 1 ≤ s < n. a ν z ν is a polynomial of degree n having all its zeros on |z| = k, k ≤ 1, then for 1 ≤ s < n, Proof. The proof of this corollary follows on the same lines as that of the theorem, but instead of applying inequality (2.8) of Lemma 2.6, we apply Lemma 2.9. We omit the details.
For s = 1 Corollary 3.3 reduces to Theorem 1.2. Further, it is clear from inequality (3.4) of Remark 3.2 that, the bound obtained from Theorem 3.1 is better than the bound obtained from Corollary 3.3 by involving some coefficients of p(z). Remark 3.4. As mentioned earlier, the bound as given by the above theorem is much better than the bound obtained by Corollary 3.3. We illustrate this by means of the following example.