#### 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 ). Let be a polynomial of degree Then according to the well-known result known as Bernstein’s inequality, we haveThe equality holds if and only if has all its zeros at the origin.

For the class of polynomials having no zero in , Lax  proved that if is a polynomial of degree having no zero in , thenThe result is best possible and equality holds for .

On the other hand, Turán  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  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.  proved the following result.

Theorem 1 (see ). 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 byThe polynomial is of degree at most and it generalizes the ordinary derivative in the sense that .

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

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

In 2008, Dewan and Upadhye  proved the following theorem.

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

Furthermore, Dewan et al.  proved the following result.

Theorem 4 (see ). 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 thatApplying the relations , , and into inequality (11), we have thatLet .
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 thatBy maximum modulus principle (see , p. 128), we obtain thatand .
Combining these relations with inequality (13), we obtain thatThis inequality together with inequality (12) yieldswhere .

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

Remark 7. In case , inequality (10) reduces toThis 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 thatwhere
We apply the relations , , and into inequality (19) to obtainwhere and .
Theorem 1 implies thatThis inequality together with inequality (20) yieldswhere 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 towhere 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)).