International Journal of Mathematics and Mathematical Sciences
Volume 2018 (2018), Article ID 5034607, 4 pages
https://doi.org/10.1155/2018/5034607

Research Article

Some Bounds for the Polar Derivative of a Polynomial

Jiraphorn Somsuwan¹ and Keaitsuda Maneeruk Nakprasit²

¹Department of Mathematics, Faculty of Science, Nakhon Phanom University, Nakhon Phanom 48000, Thailand
²Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Correspondence should be addressed to Keaitsuda Maneeruk Nakprasit; kmaneeruk@hotmail.com

Received 29 September 2017; Revised 12 January 2018; Accepted 30 January 2018; Published 1 March 2018

Academic Editor: Narendra K. Govil

Copyright © 2018 Jiraphorn Somsuwan and Keaitsuda Maneeruk Nakprasit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The polar derivative of a polynomial of degree with respect to a complex number is a polynomial , denoted by . Let . For a polynomial of degree having all its zeros in , we investigate a lower bound of modulus of on . Furthermore, we present an upper bound of modulus of on for a polynomial of degree having no zero in . In particular, our results in case generalize some well-known inequalities.

1. Introduction

One of the interesting and fruitful subjects in geometry of polynomials is the geometrical relation between the modulus of a complex polynomial on a circle and the position of zeros of this polynomial inside or outside this circle. Many propositions in the area of polynomial inequalities are presented by Bernstein-type inequalities. Many results are derived from Bernstein’s inequality. We start with a result due to Bernstein (see [1]). Let be a polynomial of degree Then according to the well-known result known as Bernstein’s inequality, we have

The equality holds if and only if has all its zeros at the origin.

For the class of polynomials having no zero in , Lax [2] proved that if is a polynomial of degree having no zero in , then

The result is best possible and equality holds for .

On the other hand, Turán [3] showed that, for a polynomial having all its zeros in ,

Inequality (3) is best possible and becomes equality for polynomials which have all its zeros on .

An extension of inequality (3) was shown by Govil [4] that, for a polynomial of degree having all its zeros in ,

Inequality (4) is best possible and equality holds for a polynomial .

In 1999, Dewan et al. [5] proved the following result.

Theorem 1 (see [5]). Let , be a polynomial of degree such that has no zero in Then, for ,

Inequality (5) is best possible and becomes equality for a polynomial , where is multiple of .

The polar derivative of a polynomial of degree with respect to a complex number , denoted by , is defined by

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

In 1998, Aziz and Rather [6] established the extension of inequality (4) to a polar derivative of a polynomial.

Theorem 2 (see [6]). Let be a polynomial of degree having all its zeros in .
Then, for every real or complex number with ,

In 2008, Dewan and Upadhye [7] proved the following theorem.

Theorem 3 (see [7]). Let be a polynomial of degree having all its zeros in .
Then, for every real or complex number with ,

Furthermore, Dewan et al. [8] proved the following result.

Theorem 4 (see [8]). If , is a polynomial of degree having no zero in , then, for every real or complex number with ,

where and .

Let In this paper, we investigate a lower bound of on for a polynomial of degree having all its zeros in and an upper bound of on for a polynomial of degree having no zero in .

2. Main Results

Theorem 5. Let , be a polynomial of degree having all its zeros in Then, for every real or complex number with and ,

where .

Proof. Let Observe that has all its zeros in .
Applying Theorem 3 to , we obtain that

Applying the relations , , and into inequality (11), we have that

Let .
Since has all its zeros in , has no zero in .
Let . Then is a polynomial of degree having no zero in . Applying Theorem 1 to with , we obtain that

By maximum modulus principle (see [9], p. 128), we obtain that

and .
Combining these relations with inequality (13), we obtain that

This inequality together with inequality (12) yields

where .

Remark 6. Inequality (10) is best possible and the equality holds for when and or and .

Remark 7. In case , inequality (10) reduces to

This means that Theorem 5 generalizes Theorem 3.

Theorem 8. If , is a polynomial of degree having no zero in , then, for every real or complex number with and ,

where and .
The equality holds for a polynomial , where with .

Proof. Let Observe that has no zero in .
Applying Theorem 4 to , we obtain that

where
We apply the relations , , and into inequality (19) to obtain

where and .
Theorem 1 implies that

This inequality together with inequality (20) yields

where and .
Next, we show that the equality holds for a polynomial , where with Let with Consider ; we obtain that Observe that the right-hand side of inequality (18) becomes Furthermore, by condition of Therefore, the equality holds for a polynomial .

Remark 9. In case , inequality (18) reduces to

where and .
That is, inequality (18) generalizes inequality (9) in Theorem 4.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The first author is supported in part by Nakhon Phanom University. The second author is supported by National Research Council of Thailand and Khon Kaen University, Thailand (Grant no. kku fmis (580010)).

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