Let be a polynomial of degree n. Then according to the well-known Bernstein’s inequality [1] .

Equality holds in (1.1) if and only if has all its zeros at the origin.

If we restrict ourselves to the class of polynomials having no zero in, then inequality (1.1) can be sharpened. It was conjectured by Erdös and later verified by Lax [2] that if in, then (1.1) can be replaced by

Inequality (1.2) is best possible and equality attains for,.

Malik [3] extended (1.2) by considering the class of polynomials of degree n not vanishing in, , and proved

Qazi [4] considered a more general class of polynomials, , having no zero in, , and obtained the following, which is a generalization as well as an improvement of (1.3).

Theorem A. If, , is a polynomial of degree n having no zero in, k ≥ 1, then

Inequality (1.4) is sharp and equality holds for the polynomial where n is a multiple of.

By involving, the above theorem was improved by Dewan et al. [5] for.

Theorem B. If, , is a polynomial of degree n having no zero in, k ≥ 1, then

Inequality (1.5) is best possible for where n is a multiple of with.

Remark 1. Theorem B proved by Dewan et al. [5] seems to have a deficiency in the sense that for the corresponding result was not specified. In fact, by simple calculation, we find the result to be the equality

Let be a polynomial of degree n and be any real or complex number, then the polar derivative of, denoted by, is defined as

The polynomial is of degree at most and it generalizes the ordinary derivative of in the sense that

.

The polynomial is called by Laguerre ( [6] , p. 48) the “émanant” of, by Pólya and Szegö [7] the “derivative of with respect to the point” and by Marden ( [8] , p. 44) simply “ the polar derivative of”.

Aziz [9] extended (1.3) to the polar derivative of by showing that if has no zero in, , then for every real or complex number with,

Inequality (1.8) is best possible and equality holds for with and.

Further, by considering a more general class of polynomials, , of degree n

having no zero in, , then for every real or complex number with, it was Dewan and Singh [10] who proved the following inequality which generalizes inequality (1.8) due to Aziz [9] .

In this paper, we first extend Theorem A to polar derivative of a polynomial, which gives an improvement of (1.9). More precisely, we prove.

Theorem 1. If, , is a polynomial of degree n having no zero in, k ≥ 1, then for every real or complex number with,

Equality in (1.10) holds for with, extremal polynomial being,.

Remark 2. To prove that the bound of Theorem 1is better than that of (1.9), it is sufficient to prove that

,

i.e. equivalently,

i.e.

which is true since, , and by (2.5) of Lemma 2.3, i.e.,.

Further, if we put in Theorem 1, we get the following result which is an improvement of inequality (1.8) due to Aziz [9] .

Corollary 1. If is a polynomial of degree n having no zero in, , then for every real or complex number with,

Remark 3. Inequality (1.13) is the corresponding polar derivative version of a result proved by Govil et al. ( [11] , Inequality (10)).

Remark 4. As mentioned earlier, inequality (1.13) improves inequality (1.8) and is evident from Remark 2, for the paticularcase.

It is of interest that as an application of Theorem 1, we have been able to obtain an independent proof of a re-

sult proved by Mir and Dar ( [12] , Theorem 1), which involves and extends Theorem B to polar de-

rivative which also improves upon Theorem 1 for. In fact, we prove

Theorem 2. If, , is a polynomial of degree n having no zero in, k ≥ 1, then for every real or complex number with,

Equality occurs in (1.11) for with, extremal polynomial being,.

If we divide both sides of the above inequalities (1.11) and (1.12) by and make, we obtain the inequalities (1.5) and (1.6) respectively.

Remark 5. For, Theorem 2 gives the following

Corollary 2. If is a polynomial of degree n having no zero in, , then for every real or complex number with,

Inequality (1.14) is best possible for, with and.

Remark 6. It is obvious that Corollary 2 is an improvement of Corollary 1.

The following lemmas are required in the proofs of the theorems.

Lemma 2.1. If is a polynomial of degree n, then on,

where

.

The above lemma is a special case of a result due to Govil and Rahman [13] .

Lemma 2.2. If is a polynomial of degree n, then for every real or complex number, we have on,

Proof of Lemma 2.2. The proof of this lemma is simple and follows as a part ( [10] , proof of Theorem 1), but for the sake of completeness, we outline it. Let. Then it is easy to verify that on,

Now, for every real or complex number, the polar derivative of with respect to is

This implies on,

which completes the proof of Lemma 2.2.

Lemma 2.3. If, , is a polynomial of degree n having no zero in, k ≥ 1, then

and

Lemma 2.3 is due to Qazi ( [4] , Proof and Remark of Lemma 1).

Proof of Theorem 1. On, by Lemma 2.1, we have

and by inequality (2.4) of Lemma 2.3, we have

Combining (3.1) and (3.2), we obtain for,

,

which gives for,

.

Now, if, then multiplying both sides of the above inequality by, we get

Inequality (3.3) when combined with Lemma 2.2, gives for and,

which is equivalent to

,

from which Theorem1 follows.

Proof of Theorem 2. First, we prove inequality (1.11).

Let. Since has no zero in, k ≥ 1 the polynomial has no zero in, k ≥ 1, for every real or complex number with. The claim is obvious if has a zero on for then and hence. If has no zero on, then we have

on and the claim follows from Rouché’s theorem. Thus, in any case has no zero in, k ≥ 1 and therefore on applying Theorem 1 to the polynomial, that is to

, where, we have for every real or complex number with,

which implies

Let be a point on the unit circle such that, then (3.4), in particular, gives

Now, we choose the argument of in (3.5) such that

.

Then (3.5) becomes

Finally, making in the above inequality, we obtain inequality (1.11).

For, the polynomial is simply having no zero in,. As has no

zero in, therefore. Then, , and . From these three equations, equality (1.12) follows readily.

BarchandChanam, (2015) Some Inequalities on Polar Derivative of Polynomial Having No Zero in a Disc. Advances in Pure Mathematics,05,796-803. doi: 10.4236/apm.2015.513073