Bounds on Derivatives in Compositions of Two Rational Functions with Prescribed Poles

TL;DR

This work extends Bernstein-type derivative bounds to compositions $r(s(z))$ of two rational functions with prescribed poles, by analyzing the class $\Re_{mn}$ and leveraging Blaschke products. The authors derive sharp upper and lower bounds on $|r'(s(z))|$ on the unit circle $T_1$ in terms of the zeros of $r\circ s$, the zeros of $s$, and the Blaschke product $B$, with explicit extremal functions achieving equality. The results generalize prior inequalities for $r\in\Re_n$ and special cases where $s(z)=z$, and they encompass both no-zeros-in-$D_k^-$ and no-zeros-in-$D_k^+$ scenarios, via intricate uses of Rouche's theorem and zero-counting. Practically, this provides a unified framework for controlling derivatives of rational compositions under prescribed pole/zero distributions, with tight constants and explicit extremals.

Abstract

This paper explores a class of rational functions r(s(z)) with degree mn, where s(z) is a polynomial of degree m. Inequalities are derived for rational functions with specified poles, extending and refining previous results in the eld.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9