  Motivated by the notion of multiplier Hermitian-Einstein metric of type
$\sigma$ introduced by Mabuchi, we introduce the notion of $\sigma$-extremal
K\"{a}hler metrics on compact K\"{a}hler manifolds, which generalizes Calabi's
extremal K\"{a}hler metrics. We characterize the existence of this metric in
terms of the coercivity of a certain functional on the space of K\"{a}hler
metrics to show that, on a Fano manifold, the existence of a multiplier
Hermitian-Einstein metric of type $\sigma$ implies the existence of a
$\sigma$-extremal K\"{a}hler metric.
