Topics on real and complex convexity

Abstract

We investigate the study of convex, strictly plurisubharmonic and the special class consisted of convex and strictly plurisubharmonic functions in convex domains of \(\mathbb{C}^n\), \(n\geq1\).
Let \(h:\mathbb{C}^n\rightarrow\mathbb{C}\) be pluriharmonic. We prove that \(\{b\in\mathbb{C}\;/| h+b|\;\textrm{is a convex function on}\; \mathbb{C}^n \}=\emptyset\), or \(\{\alpha\}\), or \(\mathbb{C}\), where \(\alpha\in\mathbb{C}\).
Now let \(\varphi_1, \varphi_2, \varphi_3:D\rightarrow\mathbb{C}\) be three holomorphic functions, \(D\) is a domain of \(\mathbb{C}^n\). Put \(u(z,w)=| w-\overline{\varphi_1}(z)|| w-\overline{\varphi_2}(z)|| w-\overline{\varphi_3}(z)|\), for \((z,w)\in D\times\mathbb{C}\). We prove that \(u\) is psh on \(D\times\mathbb{C}\) if and only if \((\varphi_1+\varphi_2+\varphi_3)\) and \((\varphi_1\varphi_2+\varphi_1\varphi_3+\varphi_2\varphi_3)\) are constant on \(D\), or \((\varphi_1+\varphi_2+\varphi_3)\) is non constant and \(\varphi_1=\varphi_2=\varphi_3\) on \(D\).