convex function then is plurisubharmonic. If and are plurisubharmonic functions, then the function is plurisubharmonic. If is a monotonically decreasing sequence of plurisubharmonic functions then is plurisubharmonic. Every continuous plurisubharmonic function can be obtained as the limit of a monotonically decreasing sequence of smooth plurisubharmonic functions. Moreover, this sequence can be chosen uniformly convergent. The inequality in the usual semi-continuity condition holds as equality, i.e. if is plurisubharmonic then (see limit superior and limit inferior for the definition of lim sup). Plurisubharmonic functions are subharmonic, for any Kähler metric. Therefore, plurisubharmonic functions satisfy the maximum principle, i.e. if is plurisubharmonic on the connected open domain and for some point then is constant. Applications In complex analysis, plurisubharmonic functions are used to describe pseudoconvex domains, domains of holomorphy and Stein manifolds. Oka theorem The main geometric application of the theory of plurisubharmonic functions is the famous theorem proven by Kiyoshi Oka in 1942. A continuous function is called exhaustive if the preimage is compact for all . A plurisubharmonic function f is called strongly plurisubharmonic if the form is positive, for some Kähler form on M. Theorem of Oka: Let M be a complex manifold, admitting a smooth, exhaustive, strongly plurisubharmonic function. Then M is Stein. Conversely, any Stein manifold admits such a function. References Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992. Robert C. Gunning. Introduction to Holomorphic Functions in Several Variables, Wadsworth & Brooks/Cole. Klimek, Pluripotential Theory, Clarendon Press 1992. External links Notes Subharmonic functions Several complex variables