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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.

Hoelder continuity property of composite Julia sets

Kosek M.

Bulletin of the Polish Academy of Sciences. Mathematics

1998

Vol. 46, no 4

391-399

In this paper we consider the composite Julia associated with a finite family of the proper polynomial mappings in [C^n]. We show its pluricomplex Green function is Hoelder continous. This yields in particular that the set preserves Markov's inequality.