Least harmonic majorant

Summary

The least harmonic majorant of a function is the smallest among its harmonic majorants. Let $u$ be a subharmonic function on a domain, with at least one harmonic majorant. Then $u$ has a least harmonic majorant, which can be built as a limit of Poisson integrals over an exhausting sequence of subdomains. Take a subharmonic function on a disc that is continuous up to the boundary. Its least harmonic majorant is the Poisson integral of its boundary values. It is used in the theory of Hardy spaces and, through its mirror image the greatest harmonic minorant, in the Riesz decomposition theorem.