A proof of Saitoh's conjecture for conjugate Hardy $H^{2}$ kernels

TL;DR

This paper proves Saitoh's conjecture by establishing a strict inequality between conjugate Hardy $H^{2}$ kernels and Bergman kernels on planar regions with multiple boundary components.

Contribution

It provides a proof of Saitoh's conjecture, demonstrating a fundamental inequality between two important types of kernels in complex analysis.

Findings

Established a strict inequality between conjugate Hardy $H^{2}$ kernels and Bergman kernels.

Extended the understanding of kernel relationships on planar regions with multiple boundaries.

Confirmed a long-standing conjecture in the theory of complex kernels.

Abstract

In this article, we obtain a strict inequality between the conjugate Hardy $H^{2}$ kernels and the Bergman kernels on planar regular regions with $n>1$ boundary components, which is a conjecture of Saitoh.