Subharmonic, Plurisubharmonic Can you give me two examples of Subharmonic, Plurisubharmonic? (and not Subharmonic, not Plurisubharmonic) . Then prove that your examples.
I'm looking forward to your help. Thanks.
 A: A $C^2$-function $u$ is subharmonic if and only if the matrix (the complex Hessian)
$$ \left( \frac{\partial^2 u}{\partial z_j \partial \bar z_k} \right)$$
has positive trace. It is plurisubharmonic if and only if the complex Hessian is positive semidefinite.
To find some examples, try to construct a few functions whose complex Hessian is
$$
\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\qquad
\begin{pmatrix} 2 & 0 \\ 0 & -1 \end{pmatrix}
\qquad
\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}
\qquad
$$
to get examples of a plurisubharmonic, subharmonic (but not psh) and non-subharmonic function respectively.
Suitably interpreted, the same conditions are true for non-$C^2$ functions as well.

Added later. Details for the first example: If $$u(z_1,z_2) = |z_1|^2 + |z_2|^2 = z_1\bar z_1 + z_2 \bar z_2,$$
then
$$ 
\frac{\partial u}{\partial z_1} = \bar z_1, \quad
\frac{\partial u}{\partial \bar z_1} = z_1, \quad
\frac{\partial u}{\partial z_2} = \bar z_2,  \quad\text{and}\quad 
\frac{\partial u}{\partial \bar z_2} = z_2.
$$
Hence
$$ 
\frac{\partial^2u}{\partial z_1 \partial \bar z_1} = 1, \quad
\frac{\partial^2 u}{\partial z_1\partial \bar z_2} = 0, \quad
\frac{\partial^2 u}{\partial z_2\partial \bar z_1} = 0,
\quad\text{and}\quad 
\frac{\partial^2 u}{\partial z_2 \partial \bar z_2} = 1.
$$
So this function works for the first example (plurisubharmonic and hence subharmonic). Make similar computations for the other examples.
