Uniform K-stability and Conformally Kähler, Einstein-Maxwell geometry on toric manifolds

Conformally Kahler, Einstein-Maxwell metrics and $f$-extremal metrics are generalization of canonical metrics in Kahler geometry. We introduce uniform K-stability for toric Kahler manifolds, and show that uniform K-stability is necessary condition for the existence of $f$-extremal metrics on toric manifolds. Furthermore, we show that uniform K-stability is equivalent to properness of relative K-energy.