Some Inequalities Involving the Derivative of Rational Functions

Abstract

In this paper, we establish some inequalities involving the modulus of the derivative of rational functions with prescribed poles and restricted zeros. The obtained results generalize some known inequalities for rational functions. Moreover, our results also contain certain known polynomial inequalities.

1. Introduction

The modulus of complex polynomials on a circle and the locations of zeros of these polynomials have been studied for several years. We start with a result due to Bernstein [1]. Let be the class of polynomials of degree at most . If is a polynomial of degree , then the famous result, known as Bernstein

Equality holds in (1) if and only if has all zeros at the origin.

conjectured in 1944 which Lax [2] proved by improving (1) that for polynomials of degree and having no zeros in , we have

Equality in (2) holds for

On the other hand, if is a polynomial of degree having all zeros in , then

Inequality (3) was demonstrated by [3] and equality in (3) holds for polynomials which have all its zeros on . In the literature [4–7], there are many improvements of inequalities (2) and (3). For the class of rational functions, we writewhere and are complex numbers. The product is known as Blaschke product and , when .

Let be the class of all polynomials of degree at most . Now we define by

Thus, is the set of all rational functions with poles at most and with finite limit at .

From now on, we denote , is the set of all points inside , and is the set of all points outside .

In 1995, Li et al. [8] extended Bernstein-type inequalities to a rational function by replacing by Blaschke product . They obtained the following results.

Theorem 1. If , thenfor . The inequality is sharp and equality holds for with .

Li et al. [8] also proved the following results.

Theorem 2. If and all the zeros of lie in , thenfor .

Theorem 3. If and all the zeros of lie in , thenfor .

Theorem 4. Suppose where has exactly poles and all the zeros of lie in . Then, for ,where is the number of zeros of .

An extension of Theorem 4 was shown by Wali and Shah [9].

Theorem 5. Suppose where has exactly poles and all the zeros of lie in . Then, for ,where is the number of zeros of .

2. Main Results

We use the following lemmas for proving our main results. The first lemma was shown by Dubinin [10] (see also [11]).

Lemma 6. For polynomial at each point of the circle at which , the inequalities are valid:

Lemma 7. If , then

Lemma 7 was proved by Aziz and Zargar [12]. The next lemma is due to Chan and Malik [13] and Li et al. [8].

Lemma 8. If , is a polynomial of degree having all the zeros in , thenfor .

Lemma 9. If and , thenfor .

In this paper, firstly, we obtain an inequality for the modulus of the derivative of rational functions with prescribed poles and restricted zeros.

Theorem 10. Suppose where has exactly poles and all the zeros of lie in except the zeros of order lying in the origin. Then, for ,where .

Proof. Let where and has all its zeros in .
It is easy to see thatThis implies thatHence,Since has all its zeros in , Lemma 8 implies thatAlso, Lemma 7 yieldsFrom (18), we getfor . Hence, for and using (18), we havefor . That is,for . Combining this with Lemma 9, we getfor . Therefore,for . This completes the proof.
By taking in Theorem 10, we get the next result.

Corollary 11. Suppose where has exactly poles and all the zeros of lie in . Then, we have for ,where .

By taking in Corollary 11, we get the following result.

Corollary 12. Suppose where has exactly poles and all the zeros of lie in . Then, we have for ,where is the number of zeros of .

If in Corollary 12, it follows that Corollary 12 reduces to Theorem 2.

Remark 13. If , let us define , for .
Then, and . We getwhere is the polar derivative of polynomial with respect to the point . It generalizes the ordinary derivative in the sense thatFor , we have .
Hence, for .
Now, the next result is obtained for the polar derivative.

Corollary 14. Suppose where and has all its zeros in except the zeros of order lying in the origin. Then,for .

Dividing both sides of the inequality (30) by and letting , we get the following result of Kumar and Lal [14].

Corollary 15. Suppose has all the zeros lying in except the zeros of order lying in the origin. Then, for ,where .

Theorem 16. Suppose where has the zeros of order with and has all its zeros lying in . Then,for .

Proof. Let where and is a polynomial of degree having all its zeros in .
By differentiating with respect to , we getIt is easy to see thatThis implies thatHence,Lemma 6 and Lemma 7 yieldfor . Therefore,for . That is,for This completes the proof.
By taking in Theorem 16, we obtain that Theorem 16 reduces to Theorem 5.
By taking and in Theorem 16, we obtain that Theorem 16 reduced to Corollary 2 of Wali and Shah [9].

Remark 17. From the conditions of Theorem 16, we have .
Thus, for . That is, our lower bound in Theorem 16 is better than the lower bound in Theorem 3 of Mir et al. [15].
From Theorem 16, we obtain the following results in term of polar derivative.

Corollary 18. Suppose has the zeros of order with and has all its zeros lying in . Then, for any complex number with ,for .
If in Corollary 18, we get the result below.

Corollary 19. Suppose has all its zeros lying in . Then, for any complex number with ,for .

Dividing both sides of the inequality (40) by and letting , we get the following result.

Corollary 20. Suppose has the zeros of order with and has all its zeros lying in . Then,for .

Corollary 21. Suppose where has the zeros of order and the zeros of order with and has all its zeros lying in . Then,for .

Proof. Let where and is a polynomial of degree having all its zeros in .
Let .
By differentiating with respect to , we obtainThe reverse triangle inequality implies thatApplying Theorem 16 to , we havefor .
Since , we getTherefore,for . Thus, the proof is complete.

Corollary 22. Suppose wherehas the zeros with for and has all its zeros lying in . Then,for and .

Proof. Letwhere has the zeros with for and the remaining zeros lie in .
Let and for .
A lower bound of is obtained by Theorem 16. Using the fact thatWe get a lower bound of as in Corollary 21.
Next, we can find a lower bound of for by a similar process by using a lower bound of from the previous process and the factfor . Finally, we getfor .

3. Conclusions

This paper gives an upper bound of a modulus of derivative of rational functions.where has exactly poles and all the zeros of lie in except the zeros of order lying in the origin. Moreover, we give a lower bound of a modulus of derivative of rational functions.where has the zeros with for and the remaining zeros lie in .

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

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

Acknowledgments

The first author was supported by Faculty of Science at Sriracha Campus, Kasetsart University, Thailand. The second author was supported by Faculty of Science, Nakhon Phanom University, Thailand.