Inequalities Concerning Rational Functions With Prescribed Poles

TL;DR

The paper addresses Bernstein-type inequalities for rational functions with prescribed poles by studying $r\in \Re_n$ with poles outside the unit disk and zeros confined to $T_k\cup D_{k+}$, using the Blaschke product $B(z)$ and Chebyshev norm on the unit circle. It develops generalizations and refinements of prior results, establishing sharp bounds on $\left|\frac{zr'(z)}{r(z)}+\frac{\beta}{1+k}|B'(z)|\right|$ for $z\in T_1$, and extends these to a coefficient-based formulation that avoids explicit zero data. The work also provides specialized cases ($k=1$) yielding refinements of earlier theorems and presents extremal examples such as $r(z)=\left(\frac{z+k}{z-a}\right)^n$ to demonstrate sharpness. These results broaden the geometric-function-theoretic framework for inequalities of rational functions with fixed poles, with implications for approximation theory and related domains.

Abstract

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.

Theorems & Definitions (25)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Theorem F
• Theorem 1
• Remark 1
• Corollary 1
• Remark 2