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].