1. Introduction to the Statement of Results
Let 
denote the space of all complex polynomials 
of degree n. If
, then
(1)
and
(2)
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) and (2) holds if and only if 
has all its zeros at the origin. It was shown by Frappier, Rahman and Ruscheweyh [4, Theorem 8] that if
, then
(3)
Clearly (3) represents a refinement of (1), since the maximum of 
on 
may be larger than the maximum of 
taken over 
roots of unityas is shown by the simple example
,
.
A. Aziz [5] showed that the bound in (3) can be considerably improved. In fact proved that if
, then for every given real
,
(4)
where
(5)
and 
is obtained by replacing 
by
. The result is best possible and equality in (4) holds for
.
Clearly inequality (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 
having no zero in
, then the inequality (1) can be sharpened. In fact, P. Erdös conjectured and later P. D. Lax [6] (see also [7]) verified that if 
for
, then (1) can be replaced by
(6)
In this connection A. Aziz [5], improved the inequality (4) by showing that if 
and 
does not vanish in
, then for every real
,
(7)
where 
is defined by (5). The result is best possible and equality in (7) holds for
.
A. Aziz [5] also proved that if 
and 
in
, then for every real 
and
,
(8)
In this paper, we first present the following result which is a refinement of inequality (7).
Theorem 1. If
, 
does not vanish in
and
, then for every real
,
(9)
where 
is defined by (5). The result is best possible and equality in (9) holds for
.
As an application of Theorem 1, we mention the corresponding improvement of (8).
Theorem 2. If
, and 
for 
and 
then for every real 
and
,
(10)
where 
is defined by (5). The result is best possible and equality in (10) holds for
.
Here we also consider the class of polynomials 
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).
Theorem 3. If 
does not vanish in
, 
and
, then for every real
,
(11)
where 
is defined by (5).
Next result is a corresponding generalization of the inequality (10).
Theorem 4. If 
does not vanish in
, 
and
, then for every real 
and
,
(12)
where 
is defined by (5).
Remark 1. For
, Theorem 3 and Theorem 4 reduces to the Theorem 1 and Theorem 2 respectively.
For the case
, we have been able to prove:
Theorem 5. If
, 
has no zero in
, 
and
, then for every real
,
(13)
provided 
and 
attain maximum at the same point on 
where
. The result is best possible and equality in (13) holds for
.
Theorem 6. If
, 
has no zero in
,
and
, then for every real 
and
,
(14)
provided 
and 
attain maximum at the same point on 
where
. The result is best possible and equality in (14) holds for
.
2. Lemmas
For the proofs of these theorems, we need the following lemmas. The first Lemma is due to A. Aziz [5].
Lemma 1. If
, then for 
and for every real
,
(15)
where 
is defined by (5).
Lemma 2. If 
and 
for
, 
, then for
,
where
.
Lemma 2 is a special cases of a result due to A. Aziz and N. A. Rather [8, Lemma 5].
Lemma 3. If 
does not vanish in
, 
, then
where
.
This Lemma is due to N. K. Govil [9].
Lemma 4. If 
is a polynomial of degree n which does not vanish in
, 
, then for 
where
.
Proof of Lemma 4. Let
. If 
has a zero on
, then 
and the result follows from Lemma 3. Henceforth we assume that 
has no zero on
, therefore 
and
If 
is any real or complex number with
, then for
,
By Rouche’s Theorem, it follows that the polynomial 
does not vanish in
, for every real or complex number 
with
. Applying Lemma 3 to the polynomial
, we get
(16)
where
Replacing 
by 
and 
by
, we obtain from (16) for
,
(17)
Now choosing the argument of 
in the left hand side of (17) such that
we obtain for
,
Letting
, we get the desired result. This proves Lemma 4.
3. Proof of the Theorems
Proof of Theorem 1. By hypothesis 
does not vanish in 
and
, therefore, by Lemma 2 with
, we have
This gives with the help of Lemma 1
Since
it follows that
which implies for 
and hence
This completes the proof of Theorem 1.
Proof of Theorem 2. Applying (2) to the polynomial 
which is of degree 
and using Theorem 1, we obtain for 
and
,
Hence for each 
and
, we have
This implies for 
and
,
which proves Theorem 2.
The proof of the Theorem 3 and 4 follows on the same lines as that of Theorems 1 and 2, so we omit the details.
Proof of Theorem 5. Since all the zeros of 
lie in
, where
, 
, by Lemma 4, we have
(18)
where
. Also by hypothesis 
and 
become maximum at the same point on
, if
(19)
then
(20)
and it can be easily verified that
Therefore, by Lemma 1
This gives with the help of (18), (19) and (20) that
which implies,
Equivalently,
and hence
This completes the proof of Theorem 5.
Theorem 6 follows on the same lines as that of Theorem 2, so we omit the details.