On real and complex convexity

Abstract

We show that the holomorphic differential equation k (k + c) = γ(k) 2 is fundamental for the study of a special class of convex and strictly plurisubharmonic functions (k : C → C be holomorphic and γ, c ∈ C). We characterize all the 4 holomorphic non-constant functions F 1 , F 2 : C → C and g 1 , g 2 : C n → C such that the function u is convex on C n × C, where u(z, w) = |F 1 (w) − g 1 (z)| 2 + |F 2 (w) − g 2 (z)| 2 , (z, w) ∈ C n × C.

Topics on real and complex convexity

“…Note that f is holomorphic on C. In fact f (z) is independent of z. We have for example, f 1 (z) does not depend only in the variable z, f 2 does not depend only in the variable z, but f 1 and f 2 are linearly independent on C. Now by the following proposition and the paper [4], we obtain an answer of the problem of the characterization of all real valued pluriharmonic functions…”

“…In this case we solve the holomorphic differential equation g ′′ (g + c) = γ(g ′ ) 2 , g is nonconstant. From the paper Abidi (II) Find all the analytic functions g : C → C such that g (4) (g ′′ + c) = γ(g (3) ) 2 , where c ∈ C and (γ ∈ C is to describe exactly). Prove that |g (3) | is convex on C.…”

“…The aim of Section 5 is to extend some results concerning psh and strictly psh functions discussed in section 4.…”

“…The aim of Section 9 is to establish theorem 9.1 and some related topics. Some good references for the study of convex functions and their variations in complex convex domains are [8,5,4,2,11]. For the study of holomorphic functions we cite the references [9,10,11,14].…”

“…is strictly psh on C 2 , then |w − g| 2 is strictly psh on C 2 , where g : C → C is holomorphic. Moreover, strictly plurisubharmonic functions plays a fundamental role in the theory of holomorphic or antiholomorphic partial differential equations and some interesting results in a slightly different direction are obtained in [3] and [4].…”