plurisubharmonic function
PlurisubharmonicFunction
2004-07-23 16:16:17
2006-06-18 21:00:35
Definition
plurisuperharmonic function
pseudoconvex function
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\begin{defn}
Let $f \colon G \subset {\mathbb{C}}^n \to {\mathbb{R}}$ be an upper semi-continuous function. $f$ is called {\em plurisubharmonic}
if for every complex line $\{ a + b z \mid z \in {\mathbb{C}} \}$
the function $z \mapsto f(a + bz)$ is a subharmonic function on the set
$\{ z \in {\mathbb{C}} \mid a + b z \in G \}$.
\end{defn}
Similarly, we could also define a {\em plurisuperharmonic} function just like
we have a superharmonic function, but again it just means that $-f$ is
plurisubharmonic, and so this extra \PMlinkescapetext{term} is not very useful.
\begin{defn}
A continuous plurisubharmonic function is said to be a {\em pseudoconvex function}.
\end{defn}
Note that since plurisubharmonic is a long word, many authors abbreviate
with {\em psh}, {\em plsh}, or {\em plush}.
\begin{thebibliography}{9}
\bibitem{Krantz:several}
Steven~G.\@ Krantz.
{\em \PMlinkescapetext{Function Theory of Several Complex Variables}},
AMS Chelsea Publishing, Providence, Rhode Island, 1992.
\end{thebibliography}