  We study a class of Hermitian metrics on complex manifolds, recently
introduced by J. Fu, Z. Wang and D. Wu, which are a generalization of Gauduchon
metrics. This class includes the one of Hermitian metrics for which the
associated fundamental 2-form is $\partial \bar\partial$-closed. Examples are
given on nilmanifolds, on products of Sasakian manifolds, on $S^1$-bundles and
via the twist construction introduced by A. Swann.
