Topics on real and complex convexity

Abidi, Jamel

Serdica Mathematical Journal, Tome 48 (2022) no. 3, pp. 149-210.

Résumé

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 \(D\).

Détail

Keywords: holomorphic, convex, plurisubharmonic functions, harmonic, holomorphic partial differential equation, complex structure, inequalities, strictly, maximum principle, 32A10, 32A60, 32U05, 32U15, 32W50