If
is a polynomial of degree
, then it is well known that
The above inequality, which is an immediate consequence of Bernstein's inequality applied to the derivative of a trigonometric polynomial, is best possible with equality holding if and only if
has all its zeros at the origin. If
in
, then
Inequality (1.2) was conjectured by Erdös and later proved by Lax [1]. If the polynomial
of degree
has all its zeros in
, then it was proved by Turán [2] that
Inequality (1.2) was generalized by Malik [3] who proved that if
in
, then
For the class of polynomials having all its zeros in
, Govil [4] proved that
Inequality (1.5) is sharp and equality holds for
. By considering a more general class of polynomials
, not vanishing in
,
, Gardner et al. [5] proved that
where
and 
Let
denote the polar derivative of the polynomial
of degree
with respect to the point
. Then
The polynomial
is of degree at most
and it generalizes the ordinary derivative in the sense that
As an extension of (1.5), it was shown by Aziz and Rather [6] that if
has all its zeros in
,
, then for
,
Inequality (1.9) was later sharpened by Dewan and Upadhye [7], who proved the following theorem.
Theorem 1 A.
Let
be a polynomial of degree
having all its zeros in
,
, then for
,
where 
Recently, Dewan et al. [8] extented inequality (1.6) to the polar derivative of a polynomial and obtained the following result.
Theorem 1 B.
If
is a polynomial of degree
having no zeros in
, then for
,
where
and 
In this paper, we will first generalize Theorem A as well as improve upon the bound obtained in inequality (1.10) by involving some of the coefficients of
. More precisely, we prove the following.
Theorem 1.1.
If
is a polynomial of degree
having all its zeros in
, then for
,
for
and
for
, where
.
Now it is easy to verify that if
, then
,
and
for
. Hence for polynomial of degree
, Theorem 1.1 is a refinement of Theorem A.
Dividing both sides of inequalities (1.12) and (1.13) by
and letting
, we get the following result.
Corollary 1.2.
If
is a polynomial of degree
having all its zeros in
,
, then
for
and
for
, where
.
These inequalities are sharp and equality holds for the polynomial
.
If we take
in the previous Theorem, we get a result, which was proved by Aziz and Dawood [9].
Next we consider a class of polynomial having no zeros in
, where
and prove the following generalization of Theorem B.
Theorem 1.3.
If
is a polynomial of degree
having no zeros in
, then for
and
,
where
Remark 1.4.
For
Theorem 1.3 reduces to Theorem B.
Remark 1.5.
Dividing the two sides of (1.16) by
and letting
, we obtain a result of Chanam and Dewan [10].