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## Robin functions and extremal functions

### Volume 80 / 2003

Annales Polonici Mathematici 80 (2003), 55-84 MSC: 31C15, 32F05. DOI: 10.4064/ap80-0-4

#### Abstract

Given a compact set $K\subset {\mathbb C}^N$, for each positive integer $n$, let \eqalign{ V^{(n)}(z)={}&V^{(n)}_K(z)\cr :={}&\sup\left\{ {1\over \hbox{deg}\,p}\,V_{p(K)}(p(z)): p \ \hbox{holomorphic polynomial}{,} \, 1\leq \hbox{deg}\,p \leq n\right\}.\cr} These “extremal-like” functions $V^{(n)}_K$ are essentially one-variable in nature and always increase to the “true” several-variable (Siciak) extremal function, $$V_K(z):=\max\left [0,\sup \left\{{1\over \hbox{deg}\,p}\log {|p(z)|}: p \ \hbox{holomorphic polynomial}{,} \, \|p\|_K\leq 1\right\}\right].$$ Our main result is that if $K$ is regular, then all of the functions $V^{(n)}_K$ are continuous; and their associated Robin functions $$\varrho_{V^{(n)}_K}(z):=\limsup_{|\lambda|\to \infty} [{V^{(n)}_K}(\lambda z)-\log(|\lambda|)]$$ increase to $\varrho_K:=\varrho_{V_K}$ for all $z$ outside a pluripolar set.

#### Authors

• T. BloomDepartment of Mathematics
University of Toronto
e-mail
• N. LevenbergUniversity of Auckland
Private Bag 92019
Auckland, New Zealand
e-mail
• S. Ma'uUniversity of Auckland
Private Bag 92019
Auckland, New Zealand
e-mail

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