A short proof of Hara and Nakai's theorem

TL;DR

This paper provides a concise proof of a theorem by Hara and Nakai, establishing an upper bound for the corona constant of finitely bordered Riemann surfaces based solely on their genus and boundary components.

Contribution

The paper introduces a shorter, more efficient proof of Hara and Nakai's theorem on corona constants for finitely bordered Riemann surfaces.

Findings

Established an upper bound for the corona constant based on genus and boundary components

Simplified the proof of Hara and Nakai's theorem

Enhanced understanding of the relationship between surface topology and corona constants

Abstract

We give a short proof of the following theorem of Hara and Nakai: for a finitely bordered Riemann surface $R$, one can find an upper bound of the corona constant of $R$ that depends only on the genus and the number of boundary components of $R$.