Inequalities Concerning Rational Functions With Prescribed Poles

Let $\Re_n$ be the set of all rational functions of the type $r(z) = p(z)/w(z),$ where $p(z)$ is a polynomial of degree at most $n$ and $w(z) = \prod_{j=1}^{n}(z-a_j)$, $|a_j|>1$ for $1\leq j\leq n$. In this paper, we set up some results for rational functions with fixed poles and restricted zeros. The obtained results bring forth generalizations and refinements of some known inequalities for rational functions and in turn produce generalizations and refinements of some polynomial inequalities as well.

💡 Research Summary

The paper investigates sharp Bernstein‑type inequalities for rational functions whose poles are prescribed outside the unit disk. Let ℜₙ denote the class of rational functions r(z)=p(z)/w(z) where p is a polynomial of degree ≤ n and w(z)=∏{j=1}^n (z−a_j) with |a_j|>1, so the poles a_j lie in the exterior of the unit circle. The associated Blaschke‑type product B(z)=∏{j=1}^n (1−a_j z)/(z−a_j) satisfies |B(z)|=1 on the unit circle and plays a central role in the estimates.

The authors begin by recalling classical results: Bernstein’s inequality ‖p′‖≤n‖p‖ for any polynomial p, the Erdős‑Lax improvement ‖p′‖≤(n/2)‖p‖ when p has no zeros in the open unit disk, and Malik’s generalization ‖p′‖≤n/(1+k)‖p‖ for polynomials whose zeros lie outside the disk of radius k≥1. For rational functions with prescribed poles, Li, Mohapatra and Rodríguez proved |r′(z)|≤|B′(z)|‖r‖ on the unit circle, and later Aziz, Zargar, and others obtained refinements when the zeros of r are restricted to certain regions (e.g., T₁∪D₁⁺).

The main contribution is Theorem 1, which provides a new, more precise upper bound for |r′(z)| when all zeros of r belong to T_k∪D_k⁺ (k≥1). Introducing a complex parameter β with |β|≤1, the theorem states that for every z∈T₁ (excluding the zeros of r),

|z r′(z)/r(z) + β/(1+k)·|B′(z)|| ≤ ½