# On Chebyshev polynomials in the complex plane

## TL;DR

This paper investigates the uniform norm estimates of Chebyshev polynomials in the complex plane, providing exact bounds for certain geometric configurations and exploring properties related to uniformly perfect sets.

## Contribution

It offers new precise estimates for Chebyshev polynomials on complex sets, including quasiconformal curves and uniformly perfect subsets of the real line.

## Key findings

- Exact norm estimates for Chebyshev polynomials on quasiconformal curves
- Norm bounds for Chebyshev polynomials on uniformly perfect sets
- Extension of estimates to complex and real subsets

## Abstract

The estimates of the uniform norm of the Chebyshev polynomials associated with a compact set $K$ in the complex plane are established. These estimates are exact (up to a constant factor) in the case where $K$ consists of a finite number of quasiconformal curves or arcs. The case where $K$ is a uniformly perfect subset of the real line is also studied.