  In this paper, we combine tools in pluripotential theory and commutative
algebra to study singularity invariants of plurisubharmonic functions. We
obtain some relationships between singularity invariants of plurisubharmonic
functions and holomorphic functions. These results lead a sharp lower bound for
the log canonical threshold of a plurisubharmonic function. It simultaneously
improves the main result in Demailly and Pham (Acta. Math 212: 1-9, 2014), the
classical result due to Skoda (Bull. Soc. Math. France 100: 353-408, 1972), as
well as the lower estimate in Fernex, Ein and Mustata (Math. Res. Lett: 10
219-236, 2003) which has received crucial applications to birational geometry
in recent years. Finally, we give some applications of our results to
singularity invariants of complex spaces.
