Abstract
For a polynomial of degree , it follows from the maximum modulus theorem that . It was shown by Ankeny and Rivlin that if for , then . In 1998, Govil and Mohapatra extended the above two inequalities to rational functions, and in this paper, we study the refinements of these results of Govil and Mohapatra.
1. Introduction and Statement of Results
Let denote the set of all complex algebraic polynomials of degree at most and let be the derivative of . For a function defined on the unit circle in the complex plane , set , the Chebyshev norm of on .
Let denote the region strictly inside and be the region strictly outside . For , , letbeing the Blaschke product, and
Then, is the set of rational functions with possible poles at and having a finite limit at . Also, note that .
1.1. Definitions
(i)For polynomial , the conjugate transpose (reciprocal) of is defined by(ii)For rational function , the conjugate transpose, , of is defined by(iii)The polynomial is self-inversive if for some .(iv)The rational function is self-inversive if for some .
It is easy to verify that if and , then , and hence, . So, is self-inversive if and only if is self-inversive.
For some related results on cubic rational splines, see Abbas et al. [1, 2].
If , then it is well known that
This inequality is an immediate consequence of the maximum modulus theorem. Furthermore, if has all its zeros in , then
Inequality (6) is due to Ankeny and Rivlin [3]. Both inequalities (5) and (6) are sharp; inequality (5) becomes equality for , where , and inequality (6) becomes equality for , where .
Govil and Mohapatra [4] gave a result analogous to inequality (5), but for rational functions, it is as follows.
Theorem 1. Ifis a rational function with for , then for ,
This result is best possible and equality holds for , where .
In the same paper, Govil and Mohapatra [4] also proved a result given as follows, that is analogous to inequality (6) for rational functions.
Theorem 2. Letwith for . If all the zeros of lie in , then for ,
This result is best possible and equality holds for the rational function , where .
In this paper, we firstly present the following refinement of the above Theorem 1. Here, is a polynomial of degree at most .
Theorem 3. Ifis a rational function with , , then for ,The result is best possible and equality holds for , where .
Remark 1. It is clear that Theorem 3 sharpens Theorem 1. Also, we can use Theorem 3 to derive a sharpening form of Bernstein’s inequality for polynomials. For this, let be a polynomial of degree . Then, , and hence by Theorem 3, for ,If on is such thatthen we get from (13)Since and , we get , and therefore, from (16), we have for ,Since (17) holds for all , where , making , where , we get that, for ,We show in Lemma 2, in Section 2, that the expression on the right hand side of (18) is an increasing function of . Note that , for if , then . On applying this fact to (18), we get that, for ,which is equivalent to that for , we haveThis rate of growth result for a polynomial, which is a sharpening of Bernstein inequality, first appeared as Lemma 3 of [5].
Before we proceed to the proof of Theorem 3, we state the following result recently proved by Mir [6] and which is a refinement of Theorem 2.
Theorem 4. Let , with for . If all the zeros of lie in , then for ,
We omit the proof of this theorem since it is already proved in the paper due to Mir [6]. However, related to this, we make the following two remarks.
Remark 2. It is clear that, in case , the above Theorem 4 reduces to Theorem 2. Also, it has been claimed by Mir [6] that, in all other cases except when , it gives a bound that is sharper than the one obtainable from Theorem 2. Although this claim seems to be correct but to justify this, it is necessary to show that for , which we show as follows.
Since for , the Blaschke productis analytic in . Furthermore, on , we have ; hence, by the maximum modulus principle, we have for , which clearly impliesBut, the above is equivalent towhich impliesThe above clearly gives that for , from which the desired inequality follows since the two sets and are the same.
Remark 3. If in Theorem 4, we multiply both sides of (21) by and then make each go to infinity, we get the following result due to Aziz and Dawood [7].
Theorem 5. Let be a polynomial of degree at most . If has no zeros in , then for ,
The result is best possible and equality holds for , where .
The above Theorem 5 clearly sharpens inequality (6) in all cases except when , in which case it clearly reduces to (6).
Remark 4. It has come to our notice that, around the same time, our paper was submitted for publication, Milovanović and Mir [8] also submitted a paper containing Theorem 3. However, our proof of Theorem 3 is different than the one given in [8] because of our proof using the generalized form of Schwarz’s lemma given in Nehari ([9], p. 167) (also, see Govil et al. ([10], p. 326)) while the proof in [8] uses a lemma due to Osserman [11].
Now, we proceed with the proof of Theorem 3, and in this regard, we present the following lemmas.
2. Lemmas
The following is a well-known generalization of Schwarz’s lemma, given in Nehari ([9], p. 167) (also, see Govil et al. ([10], p. 326)).
Lemma 1. If is analytic inside and on the circle , then for ,
Lemma 2. For , and , the function is an increasing function for .
Proof. Note thatfrom which it clearly follows that for , we haveSince the above expression is positive for , the function is increasing for , as claimed.
3. Proof of Theorem 3
Sincewith for , the functionis analytic in . Therefore, by Lemma 1 we get that, for ,and since , inequality (32) is in fact equivalent to the inequality that, for ,
Since by definition , we get from (33) that, for ,which clearly gives that, for ,
It is clear from the definition of that and this, when combined with (35), gives that, for ,which is (12) and the proof of Theorem 3 is thus complete.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.