Generalized KÃ¤hler--Einstein Metrics and Energy Functionals In this paper, we consider a generalized KÃ¤hler-Einstein equation on KÃ¤hler manifold $M$. Using the twisted $\mathcal K$-energy introduced by Song and Tian, we show that the existence of generalized KÃ¤hler-Einstein metrics with semi-positive twisting $(1, 1)$-form $\theta$ is also closely related to the properness of the twisted $\mathcal K$-energy functional. Under the condition that the twisting form $\theta$ is strictly positive at a point or $M$ admits no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of generalized KÃ¤hler-Einstein metric implies a Moser-Trudinger type inequality. Keywords:complex Monge--AmpÃ¨re equation, energy functional, generalized KÃ¤hler--Einstein metric, Moser--Trudinger type inequalityCategories:53C55, 32W20