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Let be a polynomial of degree n and for a complex number , let denote the polar derivative of the polynomial with respect to . In this paper, first we extend as well as generalize the result proved by Dewan and Mir [Inter. Jour. Math. and Math. Sci., 16 (2005), 2641-2645] to polar derivative. Besides, another result due to Dewan et al. [J. Math. Anal. Appl. 269 (2002), 489-499] is also extended to polar derivative.

Let

Equality holds in (1.1) if and only if ^{ }

Inequality (1.1) can be sharpened if we restrict ourselves to the class of polynomials having no zeros in

Inequality (1.2) is the best possible and equality attains for

Malik [

As a generalization of (1.3), Bidkham and Dewan [

Equality holds in (1.4) for

Further, Dewan and Mir [

Theorem A. If

Let

The polynomial

Aziz [

Inequality (1.6) is the best possible and equality holds for

In this paper, we establish the following result, which deduces to a result giving, in turn, a generalization as well as an extension of Theorem A to polar derivative. In fact, we prove:

Theorem 1. If

The result is the best possible and equality occurs for

Remark 1. For

Also, for

Corollary 1. If

It is seen that Corollary 1 is a generalization as well as an extension of a result due to Dewan and Mir [

Dividing both sides of (1.9) by

Corollary 2. If

The result is the best possible and the extremal polynomial is

Remark 2. Both the inequalities (1.7) and (1.9) of Theorem 1 and Corollary 1, respectively reduce to inequality (1.6) for

Further, it was shown by Turán [

The result is sharp and equality in (1.11) holds if all the zeros

As an extension of (1.11), Malik [

whereas, if

Both the estimates (1.12) and (1.13) are sharp. Equality in (1.12) holds for

Although the above result is sharp but still it is easy to see that it has two drawbacks. Firstly, the bound in (1.13) depends only on the zero of largest modulus and not on other zeros even if some of them are very close to the origin. Secondly, since the extremal polynomial in (1.13) is

bound for the polynomials

Theorem B. If

and

The result is the best possible and equality in (1.14) and (1.15) holds for

Aziz and Rather [

The result is sharp and equality holds for

While, the corresponding extension which was also a generalization of (1.13) for

Next, we further prove the following theorem in which inequality (1.18) not only extends inequality (1.14) into polar derivative but is also a generalization, while inequality (1.19) extends inequality (1.15) into polar derivative.

Theorem 2. If

and

If we divide both sides of (1.18) and (1.19) by

Remark 3. For polynomials of degree

Since

Corollary 3. If

and

Remark 4. For

and

are always non-negative so that for polynomials of degree

We require the following lemmas for the proofs of the theorems.

Lemma 2.1. If

The above result is due to Govil et al. [

Lemma 2.2. If

There is equality in (2.2) for

Lemma 2.2 is due to Jain [

Lemma 2.3. If

is a non-decreasing function of t in

Proof of Lemma 2.3.We prove this by derivative test. Now, we have

which is non-negative since

Lemma 2.4. If

Inequality (2.3) is the best possible for

Remark 5. Lemma 2.4 is of independent interest because by employing the simple fact that

of Remark 1, it gives a result which extends the theorem due to Dewan and Kaur [

The proof of Lemma 2.4 follows on the same lines as that of Lemma 2.3 due to Dewan and Mir [

Proof of Lemma 2.4. Since

which implies

Now, for

which implies on using (2.2) of Lemma 2.2,

which gives for

For

Using (2.6) to (2.5), we have

which completes the proof of Lemma 2.4.

Lemma 2.5. If

and

Lemma 2.5 is due to Dewan et al. [

Lemma 2.6. If

and

The result is sharp and equality in (2.9) and (2.10) holds for

This result is also due to Dewan et al. [

Proof of Theorem 1. Since the polynomial

or

which is equivalent to

For

hence the proof of Theorem 1 is completed.

Proof of Theorem 2. We first prove inequality (1.8). Since the zeros of

or

which is equivalent to

or

Since the polynomial

Combining (3.2) and (3.3), we get

Let

which is equivalent to

which gives

Combining (3.4) and (3.5), we get

which on simplification yields

which proves inequality (1.18) completely.

The proof of inequality (1.19) follows on the same lines as that of (1.18), but instead of applying (2.7) of Lemma 2.5 and (2.9) of Lemma 2.6, inequalities (2.8) and (2.10) respectively of Lemmas 2.5 and 2.6 are used.

BarchandChanam, (2015) Polar Derivative Versions of Polynomial Inequalities. Advances in Pure Mathematics,05,745-755. doi: 10.4236/apm.2015.512068