Narasimhans book explains that usc functions are decreasing limits of continuous functions. The passage of HÃ¶rmander's book attached is a shorter proof of characterization of pseudoconvexity, as well as a different proof that Levi pseudoconvexity is a manifestation of this (this approach is left as an exercise in Range).

I am adding to the documents the book of Krantz, "Function theory of several complex variables", 2nd ed for several reasons. I think he does a great job at explaining the real/complex tangent spaces, differentiable characterization of convexity, and Real/complex hessian. (Lecutres 10,11)

Second, I like his proof of Narasimhan's lemma best (Lectures 12,13)

Finally if you disliked my clumsy argument that all domains in C are holomorphically convex, you may find a cleaner proof in Proposition 3.1.12.However the results that we take from here are also presented in different order and spirit in Range, Chapter II.2....