Some Inequalities for the Maximum Modulus of Rational Functions

For a polynomial p(z) of degree n, it follows from the maximum modulus theorem that max|z|�R≥1|p(z)|≤Rmax|z|�1|p(z)|. It was shown by Ankeny and Rivlin that if p(z)≠ 0 for |z|< 1, thenmax|z|�R≥1|p(z)|≤ ((R + 1)/2)max|z|�1|p(z)|. 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.


Introduction and Statement of Results
Let P n denote the set of all complex algebraic polynomials p of degree at most n and let p ′ be the derivative of p. For a function f defined on the unit circle T � z||z| � 1 { } in the complex plane C, set ‖f‖ � sup z∈T |f(z)|, the Chebyshev norm of f on T.
Let D − denote the region strictly inside T and D + be the region strictly outside T. For a v ∈ C, v � 1, 2, . . . , n, let being the Blaschke product, and R n � R n a 1 , a 2 , . . . , a n � p(z) w(z) |p ∈ P n . (2) en, R n is the set of rational functions with possible poles at a 1 , a 2 , . . . , a n and having a finite limit at ∞. Also, note that B(z) ∈ R n .

Definitions
(ii) For rational function r(z) � p(z)/w(z) ∈ R n , the conjugate transpose, r * , of r is defined by It is easy to verify that if r ∈ R n and r � p/w, then r * � p * /w, and hence, r * ∈ R n . So, p/w is self-inversive if and only if p is self-inversive.
For some related results on cubic rational splines, see Abbas et al. [1,2].
If p ∈ P n , then it is well known that is inequality is an immediate consequence of the maximum modulus theorem. Furthermore, if p ∈ P n has all its zeros in T ∪ D + , then Inequality (6) is due to Ankeny and Rivlin [3]. Both inequalities (5) and (6) are sharp; inequality (5) becomes equality for p(z) � λz n , where λ ∈ C, and inequality (6) becomes equality for p(z) � αz n + β, where |α| � |β|.
Govil and Mohapatra [4] gave a result analogous to inequality (5), but for rational functions, it is as follows.
is a rational function with |a v | > 1 for 1 ≤ v ≤ n, then for |z| ≥ 1, is result is best possible and equality holds for where λ ∈ C. 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. Let
with |a v | > 1 for 1 ≤ v ≤ n. If all the zeros of r lie in T ∪ D + , then for |z| ≥ 1, is result is best possible and equality holds for the rational function r(z) � αB(z) + β, where |α| � |β|.
In this paper, we firstly present the following refinement of the above eorem 1. Here, p(z) � n v�0 α v z v is a polynomial of degree at most n.
is a rational function with e result is best possible and equality holds for Remark 1. It is clear that eorem 3 sharpens eorem 1. Also, we can use eorem 3 to derive a sharpening form of Bernstein's inequality for polynomials. For this, let p(z) � n v�0 α v z v be a polynomial of degree n. en, r(z) � p(z)/ n v�1 (z − a v ) ∈ R n , and hence by eorem 3, for |z| ≥ 1, If z * on |z| � 1 is such that then we get from (13) Since p(z) � n v�0 α v z v and r * (z) � p * (z)/ n v�1 (z − a v ), we get |r * (0)| � |α n |/ n v�1 |a v |, and therefore, from (16), we have for |z| > 1, 2 International Journal of Mathematics and Mathematical Sciences (17)

Since (17) holds for all |a
We show in Lemma 2, in Section 2, that the expression on the right hand side of (18) is an increasing function of |p(z * )|. Note that |p(z * )| ≠ 0, for if |p(z * )| � 0, then |r(z * )| ≠ ‖r‖. On applying this fact to (18), we get that, for |z| ≥ 1, which is equivalent to that for |z| � R ≥ 1, we have is 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 eorem 3, we state the following result recently proved by Mir [6] and which is a refinement of eorem 2.
If all the zeros of r lie in T ∪ D + , then for |z| ≥ 1, 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 min |z|�1 |r(z)| � 0, the above eorem 4 reduces to eorem 2. Also, it has been claimed by Mir [6] that, in all other cases except when min |z|�1 |r(z)| � 0, it gives a bound that is sharper than the one obtainable from eorem 2. Although this claim seems to be correct but to justify this, it is necessary to show that |B(z)| ≥ 1 for |z| ≥ 1, which we show as follows.
Remark 3. If in eorem 4, we multiply both sides of (21) by n v�1 a v and then make each a v go to infinity, we get the following result due to Aziz and Dawood [7]. (26) e result is best possible and equality holds for p(z) � αz n + β, where |α| � |β|. e above eorem 5 clearly sharpens inequality (6) in all cases except when min |z|�1 |p(z)| � 0, 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 eorem 3. However, our proof of eorem 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 eorem 3, and in this regard, we present the following lemmas.

Lemma 1.
If f is analytic inside and on the circle |z| � 1, then for |z| ≤ 1,