Subharmonic, Plurisubharmonic

Question

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.

Answer 1

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.

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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.