# A submultiplicative property of the Carath\'eodory metric on planar domains

## TL;DR

This paper establishes a submultiplicative inequality for the Carathéodory metric on planar domains, relating the metrics on individual domains, their intersection, and union, enhancing understanding of their geometric properties.

## Contribution

It introduces a novel inequality connecting the Carathéodory metrics on intersecting and union domains in the plane, providing new insights into their geometric relationships.

## Key findings

- Derived a submultiplicative inequality for the Carathéodory metric
- Connected metrics on domains, intersections, and unions in the plane
- Enhanced understanding of the geometric behavior of the Carathéodory metric

## Abstract

Given a pair of smoothly bounded domains $D_1, D_2 \subset \mathbb C$, the purpose of this note is to obtain an inequality that relates the Carath\'{e}odory metrics on $D_1, D_2, D_1 \cap D_2$ and $D_1 \cup D_2$.