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 .