Mirosław Baran, Leokadia Białas-Cież
Comparison principles for compact sets in $\mathbb C^N$ with HCP and Markov properties
If $\Vert z\Vert$ is a given norm in $\mathbb C^N$ and $f:\mathbb C^N\longrightarrow [-\infty,\infty)$ is a plurisubharmonic function then we can consider its growth function
$$
M_f(z)=\sup_{\Vert w\Vert\leq\Vert z\Vert}f(w).
$$
Some properties (e.g. convexity) of $M_f(z)$ are of special interest in the study of psh functions. A similar construction can be applied to the Siciak extremal function $\Phi(E,z)$ associated to a compact set $E\subset\mathbb{C}^N$. In this fashion we can obtain new comparison principles related to (HCP) and Markov property of the set $E$. We shall also prove that $L$-capacity in $\mathbb C^N$, which is a generalization of logarithmic capacity in $\mathbb C$ and is defined as
$$
C(E)=\liminf\limits_{\Vert z\Vert\rightarrow\infty}\frac{\Vert z\Vert}{\Phi^*(E,z)},\quad\Vert z\Vert=(\vert z_1\vert^2+\dots\vert z_N\vert^2)^{1/2},
$$
has the following product property
$$
C(E_1\times E_2)=\min (C(E_1),C(E_2)),\quad E_1\subset\mathbb C^{N_1},\ E_2\subset\mathbb C^{N_2}.
$$