Inequalities For The Growth Of Rational Functions With Prescribed Poles

Inequality ( 1) is a simple consequence of Maximum Modulus Principle (see [6], [7], [11]). The reverse analogue of inequality (1) whenever ν ≤ 1 was given by Varga [13], and he proved that if f ∈ P n , then

whenever 0 ≤ η ≤ 1. The equality in (1) and (2) holds whenever f (z) = λz n , λ ̸ = 0.

For the class of polynomials having no zeros inside the unit circle, T. J. Rivlin [12] proved the following result:

Theorem A. If f ∈ P n does not vanish in |z| < 1, then for 0 ≤ η ≤ 1 and |z| = 1,

The result is best possible and equality holds for f (z) = (z + ζ) n , |ζ| = 1.

As a generalization of inequality (3), A. Aziz [1] established the following result:

The result is sharp and equality holds for f (z) = (z + k) n .

Kumar and Milovanovic [4] sharpened the inequalities (3) and ( 4) by involving some of the coefficients of underlying polynomial and obtained the following result:

In the same paper, they generalized Theorem C for the class of polynomials having no zeros in |z| < k, k ≥ 1, by proving the following result:

The result is sharp and equality holds for f (z) = (z + k) n and also for f (z) = z + γ for any γ with |γ| ≥ k.

Recently, Dhankhar and Kumar [3] improved Theorems C and D, thereby sharpening Theorem B, and established the following result:

α j z j is a polynomial of degree n having no zeros in |z| < 1, then for 0 ≤ η ≤ 1

The result is sharp and equality holds for f (z) = (a + bz) n with |a| = |b| = 1 and also for f (z) = z + a for any a with |a| ≥ 1.

In the same paper, they generalized Theorem E and proved the following result:

α j z j is a polynomial of degree n having no zeros in |z| < k, k ≥ 1, then for

The result is sharp and equality holds for f (z) = (z + k) n and also for f (z) = z + γ for any γ with |γ| ≥ k.

For β j ∈ C, j = 1, 2, . . . , n, we define

Then R n is the set of all rational functions with poles β j , j = 1, 2, . . . , n at most and with finite limit at infinity. It is clear that B(z) ∈ R n and |B(z)| = 1 for |z| = 1. Throughout this paper, we shall assume that all the poles β j , j = 1, 2, . . . , n lie in |z| > 1.

The problem concerning estimation of the inequalities for the rational functions has been evolved subsequently over the last many years. Li, Mohapatra and Rodriguez [5] were the first mathematicians who obtained Bernstein-type inequalities for rational functions. For the latest publications concerning to the growth estimates for the rational functions, one can refer the papers [2], [8] and [14]. Recently Rather et al.

[10] extended the inequalities (3) and ( 4) to the rational functions and they proved the following result:

In the same paper they generalized Theorem G, which is also an extension of Theorem B to the rational functions and proved the following result:

Again Rather et al. [9] extended Theorem C to the rational functions, which is also the refinement of Theorem G and proved the following result:

In the same paper, they generalized the Theorem I, which is also the refinement of Theorem H and proved the following result:

In this section, we establish some results concerning to the rational functions of the type r(z) = f (z)/w(z), where f (z) = n j=1 α j z j and w(z) = n j=1 (z -β j ), |β j | > 1 for 1 ≤ j ≤ n by involving some coefficients of f (z). The obtained results bring forth extensions of inequalities ( 7) and ( 8) to the rational functions with prescribed poles and as a refinement of inequalities (11) and (12). We begin by presenting the following result:

Therefore, the inequality (13) sharpens the inequality (9) significantly, whenever for

whenever 0 ≤ η ≤ 1 and therefore inequality (13) sharpens the inequality (11) whenever

Remark 2. Take w(z) = (z -β) n , |β| > 1 in Theorem 1. Then inequality (13) reduces to the following inequality

Letting |β| → ∞ in inequality ( 14), we immediately get the inequality (7).

In the next result, we prove a generalization of the Theorem 1 for the class of rational functions having no zeros in |z| < k, k ≥ 1, which also sharpens the inequality (12). Therefore, for all rational functions satisfying the hypothesis of Theorem 2 except those satisfying |α 0 | = |α n |k n , our above inequality (15) sharpens the inequality (10).

Further for k ≥ 1,

whenever 0 ≤ η ≤ 1 and therefore inequality (16) sharpens the inequality ( 12) whenever |α 0 | ̸ = |α n |k n , 0 < η < 1 and n > 1.

Remark 5. Take w(z) = (z -β) n , |β| > 1 in Theorem 2. Then inequality (15) reduces to the following inequality

Letting |β| → ∞ in inequality (17), we immediately obtain inequality (8).

In order to establish our results stated above, we need the following two lemmas due to Dhankhar and Kumar [3].

Lemma 1. For any 0 ≤ η ≤ 1 and η j ≥ 1, for all j, 1 ≤ j ≤ n, we have

.

Proof of Theorem 1. By assumption r ∈ R n with no zero in |z| < 1, we have

(z -η j e iθj ), where η j ≥ 1, j = 1, 2, . . . , n. Hence, for 0 ≤ η ≤ 1 and 0 ≤ θ < 2π, we have ηe i(θ-θ