The Real and Complex Convexity
Journal Title: Journal of Mathematics and Applications - Year 2018, Vol 41, Issue
Abstract
We prove that the holomorphic differential equation [formula] (φ: C → C be a holomorphic function and (γ,c) ∈ C^2) plays a classical role on many problems of real and complex convexity. The condition exactly [formula] (independently of the constant c) is of great importance in this paper. On the other hand, let n ≥ 1, (A1,A2) ∈ C^2, and g1; g2 : C^n → C be two analytic functions. Put u(z;w) = [formula], v(z;w) = [formula], for (z;w) ∈ C^n × C. We prove that u is strictly plurisubharmonic and convex on C^n × C if and only if n = 1, (A1,A2) ∈ C^2\{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 C^n × C if and only if (A1;A2) ∈ C^2\{0}, n ∈ {1,2} and g1, g2 have several representations investigated in this paper.
Authors and Affiliations
Abidi Jamel
Keywords
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