Finite entropy vs finite energy

Probability measures with either finite Monge-Ampere energy or entropy have played a central role in recent developments Kahler geometry. In this note we make systematic study of quasi-plurisubharmonic potentials whose entropy. We show that these belong to the class ${\mathcal E}^{\frac{n}{n-1}}$, where $n$ denotes complex dimension, and provide examples showing critical exponent is sharp. Our proof relies on refined Moser-Trudinger inequalities for functions.