The theorems of Schottky and Landau for analytic functions omitting the n-th roots of unity

TL;DR

This paper establishes precise Landau- and Schottky-type theorems for analytic functions that avoid the n-th roots of unity, utilizing a sharp lower bound for the Poincaré metric on the punctured plane.

Contribution

It provides new sharp bounds for these theorems specifically for functions omitting roots of unity, advancing the understanding of their geometric properties.

Findings

Sharp Landau- and Schottky-type theorems proved

Derived a precise lower bound for the Poincaré metric

Results applicable to functions omitting roots of unity

Abstract

We prove sharp Landau- and Schottky-type theorems for analytic functions which omit the $n$-th roots of unity. The proofs are based on a sharp lower bound for the Poincar\'e metric of the complex plane punctured at the roots of unity.