Extreme values of the derivative of Blaschke products and hypergeometric polynomials

TL;DR

This paper investigates the extremal behavior of the derivative of finite Blaschke products on the unit circle, revealing that extremal cases are characterized by quotients of hypergeometric polynomials.

Contribution

It identifies classes of extremal Blaschke products that maximize or minimize the derivative difference, showing they are algebraically represented by hypergeometric polynomial quotients.

Findings

Extremal Blaschke products are quotients of hypergeometric polynomials.

Maximal and minimal derivative difference classes share the same algebraic structure.

The derivative's extremal values reflect the geometric properties of Blaschke products.

Abstract

A finite Blaschke product, restricted to the unit circle, is a smooth covering map. The maximum and minimum values of the derivative of this map reflect the geometry of the Blaschke product. We identify two classes of extremal Blaschke products: those that maximize the difference between the maximum and minimum of the derivative, and those that minimize it. Both classes turn out to have the same algebraic structure, being the quotient of two hypergeometric polynomials.