| Version current |
Version 5 |
| \begin{defn} |
\begin{defn} |
| Let $f \colon G \subset {\mathbb{C}}^n \to {\mathbb{R}}$ be an upper semi-continuous function. $f$ is called {\em plurisubharmonic} |
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 $\{ a + b z \mid z \in {\mathbb{C}} \}$ |
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 |
the function $z \mapsto f(a + bz)$ is a subharmonic function on the set |
| $\{ z \in {\mathbb{C}} \mid a + b z \in G \}$. |
$\{ z \in {\mathbb{C}} \mid a + b z \in G \}$. |
| \end{defn} |
\end{defn} |
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Similarly, we could also define a {\em plurisuperharmonic} function just like
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Similarly we could also definie a {\em plurisuperharmonic} function just like
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| we have a superharmonic function, but again it just means that $-f$ is |
we have a superharmonic function, but again it just means that $-f$ is |
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plurisubharmonic, and so this extra \PMlinkescapetext{term} is not very useful.
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plurisubharmonic and so this extra term is not very useful.
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| \begin{defn} |
\begin{defn} |
| A continuous plurisubharmonic function is said to be a {\em pseudoconvex function}. |
If $f$ is a plurisubharmonic function and further $f$ is continuous, then |
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$f$ is called a {\em pseudoconvex function}. |
| \end{defn} |
\end{defn} |
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| Note that since plurisubharmonic is a long word, many authors abbreviate |
Note that since plurisubharmonic is a long word, many authors abbreviate |
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with {\em psh}, {\em plsh}, or {\em plush}.
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with {\em psh}, {\em plsh} or {\em plush}.
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{Krantz:several} |
\bibitem{Krantz:several} |
| Steven~G.\@ Krantz. |
Steven~G.\@ Krantz. |
| {\em \PMlinkescapetext{Function Theory of Several Complex Variables}}, |
{\em \PMlinkescapetext{Function Theory of Several Complex Variables}}, |
| AMS Chelsea Publishing, Providence, Rhode Island, 1992. |
AMS Chelsea Publishing, Providence, Rhode Island, 1992. |
| \end{thebibliography} |
\end{thebibliography} |
