2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/128010Mabuchi introduced multiplier Hermitian structures on compact Kahler manifolds and defined metrics similar to Kahler-Einstein metrics under these structures. In this note we generalize the inequality of Moser-Trudinger type on Kahler-Einstein manifolds to this case.Minor change made at the end of the section 1Differential Geometrytext