#### Abstract

In this paper, we consider a class of rational functions of degree where is a polynomial of degree and establish some inequalities for rational functions with prescribed poles which generalize and refine the result of I. Qasim and A. Liman.

#### 1. Introduction

Let denote the class of all complex polynomials of degree at most and let be a positive real number. We denote , and . Consider a polynomial of degree . In 1926, Bernstein [1] presented the following well-known inequality:

Equality holds in (1) only for , where . If we restrict to the class of polynomials having no zeros in , inequality (1) can be sharpened. In fact, it was conjectured by P. Erds and later proved by Lax [2] that if has no zeros in , then

For the class of polynomials having no zeros in , Turán [3] proved that

For , we let and

The product is known as a Blaschke product.

Then, is the set of rational functions with at most poles and with finite limit at infinity. For defined on , we denote , the Chebyshev norm of on . Throughout this paper, we assume that all poles are in .

In 1995, Li et al. [4] proved some inequalities similar to (1), (2), and (3) for rational functions. Among other things, they proved the following result.

Theorem 1 (see [4]). Let with all its zeros lying in . Then, for ,Equality holds for with .

In 1997, inequality (5) was improved by Aziz and Shah [5] under the same hypothesis. They obtained the following theorem.

Theorem 2 (see [5]). Let with all its zeros lying in . Then, for ,where . Equality holds for where and is real.

In 1999, Aziz and Zarger [6] considered a class of rational functions not vanishing in , where , and established the following generalization of Theorem 1.

Theorem 3 (see [6]). Let with all its zeros lying in , where . Then, for ,Equality holds for and evaluated at , where .

Recently, inequalities (6) and (7) were improved by Arunrat and Nakprasit [7] under the same hypothesis. They obtained the following theorem.

Theorem 4 (see [7]). Let , where has exactly poles at and all its zeros lie in , . Then, for ,where is the number of zeros of with counting multiplicity and . Equality holds for and evaluated at , .

In 2015, Qasim and Liman [8] considered a class of rational functions with all poles lying in , defined bywhere is a polynomial of degree with all its zeros lying in and . Letand the Blaschke product

They proved the following generalization of inequality (5).

Theorem 5. (see [8]). Let , where has no zeros in and all zeros of lie in . Then, for ,where . The inequality is sharp and equality holds for with and .

Observe that if has a zero on , then , and we obtain a trivial inequality:

In this paper, we consider the class of rational functions having no zeros in , where , and prove the generalization of the result of Qasim and Liman [8].

#### 2. Lemmas

For the proof of our main theorems, we need the following lemmas. These two lemmas are due to Li et al. [4].

Lemma 1 (see [4]). Let . If all zeros of lie in , then, for ,where .

Lemma 2 (see [4]). If and , then, for ,Equality holds for with .

Lemma 3 is due to Aziz and Dawood [9].

Lemma 3 (see [9]). If and has all its zeros in , thenThe inequality is sharp and equality holds for polynomials having all zeros at the origin.

Lemma 4 is due to Aziz and Shah [5], and Lemma 5 is due to Arunrat and Nakprasit [7].

Lemma 4 (see [5]). If is Blaschke product and is real, , then has all its zeros in , for every .

Lemma 5 (see [7]). Assume that , where has exactly poles at . Let be the number of zeros of with counting multiplicity. If all zeros of lie in , where , and with , then

#### 3. Main Theorems

In this section, we state and prove main results. One of them generalizes the result of Qasim and Liman [8].

Theorem 6. Let with in and all zeros of lie in . Then, for ,where and .
Equality holds for where , , and is real.

Proof. Let without zeros in and .
Therefore, for . If has a zero on , then , and hence, for every with , we get that . In case has no zeros on , we have for every with that for . It follows from Rouche’s theorem that rational functions and have the same number of zeros in . That is, for every with , has no zeros in . We first assume that . Lemma 1 yields that for ,Let . Then, .
Consequently, .
Since , we have , , , and soSince is real, we obtain that .
Then,where the inequality comes from (19).
This implies that for which are not the zeros of ,where with .
Moreover, and .
Applying these relations into (22), we obtain thatfor with and every with .
Choose the argument of so thatfor with .
Substituting relation (24) into (23), we obtain thatLetting , we obtainLemma 2 implies thatThus,For with , we have .
From Lemma 3, we obtain thatIt follows from (28) thatThis proves inequality for . In case , we obtain that .
This implies that the above inequality is trivially true.
Therefore, inequality (18) holds for all .
Next, we show that equality holds for where and . Lemma 4 implies that has all its zeros in . Moreover, we obtain thatConsider .
This implies that . Then, for , .
The right side of inequality (18) isThus, this bound is best possible.

Theorem 7. Let with in , and all zeros of lie in . Then, for ,where is the number of zeros of with counting multiplicity, , and . Equality holds for , where and , and at .

Proof. Let without zeros in , where .
Let and be the number of zeros of with counting multiplicity. Therefore, for . If has a zero on , then , and hence, for every with , we obtain that . In case has no zeros on , we have for every with that for . Therefore, it follows from Rouche’s theorem that rational functions and have the same number of zeros in . That is, for every with , has no zeros in . We first assume that . Lemma 5 yields that for ,Let . Then, .
Consequently, .
Since , we have , , , and soSince is real, we obtain that .
Then,where the inequality comes from (34).
This implies that for which are not zeros of ,where with ,Moreover, and .
Applying these relations into (37), we obtain thatfor with and every with .
Choose the argument of so thatfor with .
Triangle inequality yields that
Note that which implies thatSubstituting relations (40) and (39) into (38), we obtain thatLetting , we obtainLemma 2 implies thatEquivalently,Hence,Then,That is,Thus,For with , we haveFrom Lemma 3, we obtain thatTherefore, it follows from (48) thatwhere is the number of zeros of with counting multiplicity, , and . This proves inequality for .
In case , we obtain that .
This implies that the above inequality is trivially true.
Therefore, inequality (33) holds for all .
Next, we show that equality holds for where and , , and at . First, we observe that , , and .
Then,The right side of inequality (33) isThus, this bound is best possible.
Theorem 7 simplifies to Theorem 4 when .
Observe that for all . We obtain an immediately consequence of Theorem 7 as follows.

Corollary 1. Let with in , and all zeros of lie in . Then, for ,where and . Equality holds for where and , , and at .

From Corollary 1, if has all its zeros in with at least one zero on , we obtain the following corollary.

Corollary 2. Let and in with at least one zero on , where , and all zeros of lie in . Then, for ,where . Equality holds for where and , , and at .

When and has a zero on , Corollary 2 generalizes Theorem 5.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

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

#### Acknowledgments

This work has received a scholarship under the Research Fund for Supporting Lecturer to Admit High Potential Student to Study and Research on His Expert Program Year 2018 from the Graduate School, Khon Kaen University, Thailand (Grant no. 612JT217).