Results for Self-Inversive Rational Functions

: In this paper, we ﬁnd some relations between maximum modulus of a rational function r ( z ) satisfying r ( z ) = B ( z ) r (1 /z ) and the maximum modulus of its derivative. We also ﬁnd analogue of Cohn’s Theorem for rational functions.


Introduction
Let P n denote the space of complex polynomials p(z) := B(z) is known as finite Blaschke product.Let p(z) be a polynomial of degree at most n with complex variable z.We consider the following space of rational functions Throughout this paper, we shall assume that all the poles z 1 , z 2 , . . ., z n are in D + unless otherwise stated.
For the case when all the poles are in D − , we can obtain analogous results with suitable modification of our method.

Definition of conjugate transpose
1.For p(z) := n j=0 α j z j , the conjugate transpose (reciprocal) p * of p is defined by In 1927, Bernstein [3] proved the following result.
where the equality holds for polynomials having all zeros at the origin.
In 1969, Malik [5] improved inequality (1.2) and proved the following: where As an easy consequence of inequality (1.3), we have the following result which improves inequality (1.2) for self-inversive polynomials.
For a complex number α and for p ∈ P n , let D α p(z) is a polynomial of degree at most n − 1 and is known as polar derivative of p with respect to α.It generalizes the ordinary derivative in the sense that Aziz and Shah [2] extended inequality (1.2) to the polar derivative of a polynomial and proved the following result.Theorem B. If p ∈ P n , then for every α with α ∈ T ∪ D + and z ∈ T, (1.5) Li, Mohapatra and Rodriguez [6] extended inequality (1.2) and (1.4) to rational functions with prescribed poles and proved the following results.
Regarding the number of zeros of a self-inversive polynomial inside a unit circle, we have the following well-known result [4].
Theorem E(Cohn's Theorem).Let g(z) be a self inversive polynomial, then g(z) has the same number of zeros inside the unit circle as does the polynomial c[g ′ (z)] * .
In this paper, we give improvement of inequality (1.6) for self-reciprocal rational functions.Inequality for polar derivative of a polynomial is deduced which improves inequality (1.5) for the class of polynomials p(z) satisfying p(z) = z n p (1/z).Moreover, the analogue of Cohn's Theorem for rational functions is also discussed.

Main Results
The first result gives the improvement of inequality (1.6) for self-reciprocal rational functions.
When we look at the analogous of Cohn's theorem for rational functions of the form r(z) = p(z)/w(z), we see that since r ′ (z) = [w(z)p ′ (z) − w ′ (z)p(z)] /(w(z)) 2 , therefore, the zeros of w(z) also play a role.However, one would expect that analogous of Cohn's Theorem might be true, if we restrict zeros of w(z) in a region.The feasible regions where we can restrict zeros of w(z) are either |z| < 1 or |z| > 1.But both the cases does not work as is clear from the following two examples: The following result gives the indirect analogue of Cohn's Theorem for rational functions

Lemmas
For the proofs of these theorems we need the following lemma due to Li, Mahapatra and Rodrigues [6].

Proofs of the Theorems
Proof of Theorem 1.Since where r 1 (z) and r 2 (z) are rational functions of degree less than for equal to n.Also, r(z) = B(z)r (1/z), therefore, and We claim that and To prove our claim, let where α is a complex number with |α| = 1.Then This shows that F (z) is a self-inversive rational function of degree n and therefore, by Theorem D, we have for z ∈ T Results for Self-Inversive Rational Functions