On the Błocki–Zwonek conjectures and beyond

Abstract

Let ${\Omega}$ be a bounded pseudoconvex domain in ${\mathbb{C}^n}$, and let ${g_{\Omega}(z,a)}$ be the pluricomplex Green function with pole at a in ${\Omega}$. Błocki and Zwonek conjectured that the function given by

$$\begin{array}{ll}\alpha = \alpha_{\Omega}, a: (- \infty, 0) \ni t \mapsto \alpha (t) = e^{-2nt} \lambda_n \left( \{z \in \Omega: g_{\Omega}(z, a) < t \} \right)\end{array}$$

is nondecreasing, and that the function given by

$$\begin{array}{ll}\beta = \beta_{\Omega}, a: (-\infty, 0) \ni t \to \beta(t)= \log \left(\lambda_n \left(\{z \in \Omega: g_{\Omega}(z,a)< t\}\right)\right)\end{array}$$

is convex. Here ${\lambda_{n}}$ is the Lebesgue measure in ${\mathbb{C}^n}$. In this note we give an affirmative answer to these conjectures when ${\Omega}$ is biholomorphic to a bounded, balanced, and pseudoconvex domain in ${\mathbb{C}^n}$, ${n\geq 1}$. The aim of this note is to consider generalizations of the functions ${\alpha}$, ${\beta}$ defined by the Green function with two poles in ${\mathbb{D}\subset\mathbb{C}}$. We prove that ${\alpha}$ is not nondecreasing, and ${\beta}$ is not convex. By using the product property for pluricomplex Green functions, we then generalize this to n-dimensions. Finally, we end this note by considering two other possibilities generalizing the Błocki–Zwonek conjectures.