Smale's mean value conjecture for finite Blaschke products

TL;DR

This paper explores Smale's mean value conjecture and its dual for finite Blaschke products, establishing new bounds and drawing parallels with polynomial cases to advance understanding in complex analysis.

Contribution

It introduces new results on the dual conjecture for finite Blaschke products, improving bounds previously established for polynomial cases.

Findings

Improved bounds for the dual mean value conjecture for polynomials

Established results for Smale's conjecture in the context of finite Blaschke products

Enhanced understanding of the relationship between polynomials and Blaschke products

Abstract

Motivated by a dictionary between polynomials and finite Blaschke products, we study both Smale's mean value conjecture and its dual conjecture for finite Blaschke products in this paper. Our result on the dual conjecture for finite Blaschke products allows us to improve a bound obtained by V. Dubinin and T. Sugawa for the dual mean value conjecture for polynomials.