Turan type inequalities for rational functions with pescribed poles and restricted zeros

TL;DR

This work develops Turán-type inequalities for rational functions with prescribed poles and zeros restricted to the unit disk or its inverse, by formulating a generalized lower bound for $|r'(z)|$ in terms of the Blaschke product $B(z)$ and the zeros of the numerator. The authors define the rational-function class $\Re_n$ and derive a sharp bound that incorporates the pole count, the number and location of zeros, and a possible zero of order $s$ at the origin; the result recovers known inequalities as special cases and naturally connects to polar derivatives via a limiting process. The approach relies on auxiliary lemmas relating $r$, its reflected function $r^{*}$, and $B(z)$ on $|z|=1$, together with a real-part analysis of $z r'(z)/r(z)$. Overall, the paper extends Turán-type bounds to a broader class of rational functions and provides refined, sharp inequalities that also yield corollaries for polynomials and their polar derivatives with zeros constrained to $D_k \cup D_k^{-}$.

Abstract

In this paper, we establish some inequalities for rational functions with prescribed poles having s-fold zeros at origin and also show that it implies some inequalities for polynomials and their polar derivatives.

Theorems & Definitions (19)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 3.1
• Remark 3.1
• Corollary 3.1
• Corollary 3.2
• Remark 3.2
• Corollary 3.3
• Corollary 3.4