In this note, we study the Laplacian comparison theorem and the sub-mean-value theorem for a special type of Hermitian manifolds called multiplier Hermitian manifolds. By conformal change of the metrics, this covers much wider objects than in the case of ordinary Kähler manifolds.
"Laplacian comparison and sub-mean-value theorem for multiplier Hermitian manifolds." J. Math. Soc. Japan 56 (4) 1211 - 1219, October, 2004. https://doi.org/10.2969/jmsj/1190905456