Plurisubharmonic function

Remarks

Clause 2 says that for all $a, b$ the function $\lambda \mapsto u(a + \lambda b)$ is subharmonic or identically $-\infty$ on each component of its domain. Many authors also require that $u$ is not identically $-\infty$ on any connected component of $\Omega$. A function $u$ of class $C^2$ is plurisubharmonic if and only if $$\sum_{j,k=1}^n \frac{\partial^2 u}{\partial z_j \partial \bar z_k}(z) \, w_j \bar w_k \ge 0 \quad \text{for all } z \in \Omega,\ w \in \C^n .$$ Example: $u(z) = \abs{z_1}^2 + \cdots + \abs{z_n}^2$ has complex Hessian equal to the identity matrix, which is positive definite, so $u$ is plurisubharmonic. Non-example: $u(z_1, z_2) = \abs{z_1}^2 - \abs{z_2}^2$ is smooth, but on the line $\lambda \mapsto (0, \lambda)$ it is $-\abs{\lambda}^2$, which is not subharmonic, so clause 2 fails at $a = 0$, $b = (0, 1)$. Plurisubharmonic functions are subharmonic as functions on $\R^{2n}$, but the converse fails, as the non-example shows. The abbreviation "psh" is common.