Some Properties of Holomorphic Functions having Convex Absolute Values and Applications

Abstract

 We study some properties concerning the convexity, plurisubharmonicity or other properties (such as convex and strictly psh, or convex and s trictly psh but not strictly convex) of certain special classes constructed from holomorphic functions. We prove that we have a key role between real and complex convexity in the theory of functions.  On the other hand, let g : C^n → C be a holomorphic nonconstant function. We prove that | g | is convex on C^n if and only if g has a classical holomorphic representation. Several applications of this criterion are obtained in the theory of functions consisting by (convex and strictly plurisubharmonic functions), or (convex and strictly plurisbharmonic but not strictly convex functions) and related topics.