Research Article | Open Access

Volume 2020 |Article ID 4378409 | https://doi.org/10.1155/2020/4378409

Nuttapong Arunrat, Keaitsuda Maneeruk Nakprasit, "Inequalities for the Derivative of Rational Functions with Prescribed Poles", International Journal of Mathematics and Mathematical Sciences, vol. 2020, Article ID 4378409, 7 pages, 2020. https://doi.org/10.1155/2020/4378409

# Inequalities for the Derivative of Rational Functions with Prescribed Poles

Accepted17 Aug 2020
Published15 Sep 2020

#### 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  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  that if has no zeros in , then

For the class of polynomials having no zeros in , Turán  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.  proved some inequalities similar to (1), (2), and (3) for rational functions. Among other things, they proved the following result.

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

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

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

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

Theorem 3 (see ). 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  under the same hypothesis. They obtained the following theorem.

Theorem 4 (see ). 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  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 ). 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 .

#### 2. Lemmas

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

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

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

Lemma 3 is due to Aziz and Dawood .

Lemma 3 (see ). 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 , and Lemma 5 is due to Arunrat and Nakprasit .

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

Lemma 5 (see ). 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 .

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) is