A multiplier Hermitian manifold is a quantitative generalization of a Kähler-Ricci soliton. If \((M,\omega)\) is an \(n\)-dimensional connected complete Kähler manifold the pair \((M,\widetilde \omega)\) is a multiplier Hermitian manifold, where \(\widetilde \omega =\exp (-\psi/n) \omega\) and \(\psi\) is a suitably defined function on \(M\).
The authors obtain two comparison theorems for multiplier Hermitian manifolds. The main result is a sub-mean-value property for these manifolds, and the key of the proof lies in proving a Laplacian comparison result for the same manifolds.