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Content available remote The real and complex convexity
We prove that the holomorphic differential equation ϕ’’(ϕ+c) = γ(ϕ’)² (ϕ:C→C be a holomorphic function and (γ, c) ϵ C²) plays a classical role on many problems of real and complex convexity. The condition exactly γ ϵ [wzór] (independently of the constant c) is of great importance in this paper. On the other hand, let n ≥ 1, (A₁, A₂) ϵ C² and g₁, g₂ : Cᵑ → C be two analytic functions. Put u(z, w) = │A ₁w - g₁(z) │² + │A₂w - g₂(z) │²v(z,w) = │A₁w - g₁(z) │² + │ A₂w - g₂(z) │², for (z,w) ϵ Cᵑ x C. We prove that u is strictly plurisubharmonic and convex on Cᵑ x C if and only if n = 1, (A₁, A₂) ϵ C² \{0} and the functions g₁ and g₂ have a classical representation form described in the present paper. Now v is convex and strictly psh on Cᵑ x C if and only if (A₁, A₂) ϵ C² \{0}, n ϵ {1,2} and and g₁, g₂ have several representations investigated in this paper.
Content available remote Some global solutions of the complex Monge-Ampere equation
We give some sufficient condition which guarantee that for given plurisubharmonic function u (satisfying this condition) one can solve the global Monge-Ampere equation in [C^n] : (d[d^c]w)n = d[my], for any positive Borel measures d[my], which are "close" to Monge-Ampere measure of the function u.
Content available remote Plurisubharmonic functions on Reinhardt domains in (C*)^n
We are intersting in the Minimum Principle for plurisubharmonic functions. This problem has been studied by Kiselman in the invariant case by translation. We have been inspired by this study to exhibit a new class of pseudoconvex open sets verifing the minimum principle for plurisubharmonic functions and which admit pseudoconvex projections. Then, we introduce Reinhardt domains and invariant functions by rotation.
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