Maximal Blaschke Products

TL;DR

This paper introduces maximal Blaschke products as extremal functions for maximizing derivatives at fixed points with prescribed critical points, extending classical results and establishing their fundamental properties.

Contribution

It demonstrates that extremal functions are unique indestructible Blaschke products, leading to the concept of maximal Blaschke products as a significant generalization.

Findings

Maximal Blaschke products are essentially unique extremal functions.

They extend the Nehari--Schwarz Lemma to a broader class.

Maximal Blaschke products form an infinite generalization of finite Blaschke products.

Abstract

We consider the classical problem of maximizing the derivative at a fixed point over the set of all bounded analytic functions in the unit disk with prescribed critical points. We show that the extremal function is essentially unique and always an indestructible Blaschke product. This result extends the Nehari--Schwarz Lemma and leads to a new class of Blaschke products called maximal Blaschke products. We establish a number of properties of maximal Blaschke products, which indicate that maximal Blaschke products constitute an appropriate infinite generalization of the class of finite Blaschke products.