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A contribution on real and complex convexity in several complex variables

Abidi Jamel

Journal of Mathematics and Applications

2022

Vol. 45

21--58

Let f, g : Cn → C be holomorphic functions. Define u(z, w) = |w − f (z)|4 + |w − g(z)|4, v(z, w) = |w − f (z)|2 + |w − g(z)|2, for (z, w) ∈ Cn × C. A comparison between the convexity of u and v is obtained under suitable conditions. Now consider four holomorphic functions φ1, φ2 : Cm → C and g1, g2 : Cn → C. We prove that F = |φ1 − g1|2 + |φ2 − g2|2 is strictly convex on Cn × Cm if and only if n = m = 1 and φ1, φ2, g1, g2 are affine functions with (φ′1g′2 − φ′2g′1)̸ = 0. Finally, it is shown that the product of four absolute values of pluriharmonic functions is plurisubharmonic if and only if the functions satisfy special conditions as well.

The real and complex convexity

Jamel A.

Journal of Mathematics and Applications

2018

Vol. 41

123--156

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.

Some Global Solutions of the Complex Monge-Ampère Equation

Czyż R.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 3

287-296

We give some sufficient condition which guarantee that for given plurisubharmonic function u (satisfying this condition) one can solve the global Monge-Ampère equation in Cn: (ddcv)n = dμ, for any positive Borel measures dμ, which are "close" to Monge-Ampère measure of the function u.

Plurisubharmonic functions on Reinhardt domains in (C*)^n

Derrab F.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2001

[Z] 41

69-92

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.