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)| ≤ Rn max|z|=1 |p(z)|. It was shown by Ankeny and Rivlin in 1955 that if p(z) 6= 0 for |z| < 1 then max|z|=R≥1 |p(z)| ≤ R +1 2 max|z|=1 |p(z)|. These two results were extended to rational functions by Govil and Mohapatra [4]. In this paper, we give 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 + the region strictly outside T. For a v ∈ C, v = 1, 2, . . . , n, let w(z) = n v=1 (z −a v ), B(z) = n v=1 (1−a v z)/(z −a v ) being the Blashke product, and R n = R n (a 1 , a 2 , . . . , a n ) = {p(z)/w(z) | p ∈ P n }. Then 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 .
(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. If p ∈ P n then it is well known that This inequality is an immediate consequence of the Maximum Modulus Theorem. Further, if p has all its zeros in T ∪ D + , then The inequality (1.2) is due to Ankeny and Rivlin [1]. Both inequalities (1.1) and (1.2) are sharp, inequality (1.1) becomes equality for p(z) = λz n where λ ∈ C, and inequality (1.2) becomes equality for p(z) = αz n + β where |α| = |β|. Govil and Mohapatra [4] gave a result analogous to inequality (1.1), but for rational functions, as follows.
is a rational function with |a v | > 1 for 1 ≤ v ≤ n, then for |z| ≥ 1, This result is best possible and equality holds for where λ ∈ C.
In the same paper, Govil and Mohapatra [4] also proved a result given below, that is analogous to inequality (1.2) for rational functions.

THEOREM B. Let
If all the zeros of r lie in T ∪ D + , then for |z| ≥ 1 This result is best possible and equality holds for the rational function r(z) = αB(z) + β where |α| = |β|.
In this paper we prove the following refinements of the above two theorems. Here The result is best possible and equality holds for r(z) = λB(z) where λ ∈ C..
It is clear that Theorem 1 sharpens Theorem A. Also, we can use Theorem 1 to derive a sharpening form of Bernstein's Inequality for polynomials. For this, let ∈ R n and hence by Theorem 1, for |z| ≥ 1, If z * on |z| = 1 is such that , we get |r * (0)| = |α n | n v=1 |a v | and therefore from (1.8) we have for |z| > 1, We show in Lemma 5 in the next section that (1.10) implies for |z| ≥ 1 which is equivalent to that for |z| = R ≥ 1, This rate of growth result for a polynomial, which is a sharpening of Bernstein Inequality, first appeared as Lemma 3 of [2]. As a refinement of Theorem B, we shall prove If all the zeros of r lie in T ∪ D + , then for |z| ≥ 1 Clearly Theorems 1.1 and 1.2 without any additional hypotheses, give bounds that are sharper than those obtainable from Theorems A and B respectively.

Lemmas
The following is a well known generalization of Schwarz's Lemma (see, for example, [3]).
Lemma 2.1. If f is analytic inside and on the circle |z| = 1, then for |z| ≤ 1, The next two results are due to Govil and Mohapatra [4].
Lemma 2.2. Let r ∈ R n with all its poles in D + . If r has all its zeros in T ∪ D + , then for all |z| ≥ 1, |r(z)| ≤ |r * (z)|.
Lemma 2.3. Let r ∈ R n with all its poles in D + . Then for |z| ≥ 1, Lemma 2.4. Let r ∈ R n with all its poles in D + . If r has all its zeros in T ∪ D + , then for |z| ≥ 1, we have Proof. Since the rational function r has no zeros in D − hence for every α ∈ C with |α| < 1, the rational function r(z) − α min |z|=1 |r(z)| has no zero in D − and has all its poles, like r, in D + . Applying Lemma 2.2 to r(z) − α min |z|=1 |r(z)| we get that for |z| ≥ 1 and so for |z| ≥ 1, With the appropriate choice of arg(α) we then have for |z| ≥ 1, Note that r has no zeros in D − and so is analytic in |z| ≤ 1. Hence by the Minimum Modulus Theorem, we have |r(z)| > |α| min |z|=1 |r(z)| for |z| ≤ 1. Therefore for |z| ≥ 1 we get which clearly implies that the right-hand side of (2. Lemma 2.5. The function where α n , z ∈ C with z = 0, is an increasing function for x ≥ 0. Proof. We have g (x) = |z|x 2 + 2|α n ||z| + |z| |α n | 2 (|α n | + |z|x) 2 ≥ 0 for x ≥ 0. So g is an increasing function for x ≥ 0, as claimed.