On the Derivative of a Polynomial

Certain refinements and generalizations of some well known inequalities concerning the polynomials and their derivatives are obtained.


Introduction to the Statement of Results
Let denote the space of all complex polynomials and Inequality ( 1) is an immediate consequence of S.Bernstein's theorem (see [1]) on the derivative of a trigonometric polynomial.Inequality (2) is a simple deduction from the maximum modulus principle (see [2, p. 346] or [3, p. 137]).
Both the inequalities (1) and ( 2) are sharp and the equality in (1) Aziz [5] showed that the bound in (3) can be considerably improved.In fact proved that if , then for every given real where (5) and π M   is obtained by replacing  by π   .The result is best possible and equality in (4) holds for 4) is an interesting refinement of inequality (3) and hence of Bernstein inequality (1) as well.
If we restrict ourselves to the class of polynomials n P P  having no zero in 1 z  , then the inequality (1) can be sharpened.In fact, P. Erdös conjectured and later P. D. Lax [6] (see also [7]) verified that if   0 P z  for 1 z  , then (1) can be replaced by In this connection A. Aziz [5], improved the inequality (4) by showing that if n and does not vanish in where M  is defined by (5).The result is best possible and equality in (7) holds for   n i P z z e    In this paper, we first present the following result which is a refinement of inequality (7).
where M  is defined by (5).The result is best possible and equality in (9) holds for   n i P z z e    .
As an application of Theorem 1, we mention the corresponding improvement of (8).

Theorem 2. If
, and for where M  is defined by (5).The result is best possible and equality in (10) holds for   n i P z z e    .
Here we also consider the class of polynomials n having no zero in and present some generalizations of the inequalities ( 9) and ( 10).First we consider the case and prove the following result which is a generalization of inequality (9).
where M  is defined by (5).
Next result is a corresponding generalization of the inequality (10).

Theorem 4. If does not vanish in
where M  is defined by (5).Remark 1.For , Theorem 3 and Theorem 4 reduces to the Theorem 1 and Theorem 2 respectively.

k 
For the case , we have been able to prove: The result is best possible and equality in (13) holds for .
where M  is defined by (5).Lemma 2. If n P P  and for where     This Lemma is due to N. K. Govil [9].Lemma 4. If

  P z is a polynomial of degree n which does not vani sh in z k
.
 in the l side of (17) such that eft hand   , we get the desired result.This proves Lemma 4. f the othesis does not

Proof o
Proof of Theorem 1.By hyp vanish in This gives with the help of Lemma 1 This completes the proof of Theorem 1.

Proof of
Hence for each and , we have The proof of the Theorem 3 and 4 follows on the e lines as that of Theorems 1 and 2, so we omit the de ll the zeros of hich proves Theorem sam tails.Proof of Theorem 5. Since a where . Also by hypothesis and   Q z  become maximum at the same point on an be easily verified that and it c ," Transactions of the American Mathematical Society, Vol.288, 1985, pp.69-99.doi:10.1090/S0002-9947-1985-0773048-1[5] A. Aziz, "A Refinement of an Inequality of S.Bernstein," of Theorem 5. Theorem 6 follows on the same lines as that of Theorem 2, so we omit the details.
is a special cases of a result due to A. Aziz an