In Example 7.4b, Chapter I of Demailly, he propose a Hartogs domain $\Omega=\{(z_1,z')\in \mathbb{C}\times\omega ; \mathrm{log}(\vert z_1 \vert)+u(z') < 0\}$ for upper semicontinuous function $u : \omega \subset \mathbb{C}^{n-1} \rightarrow [-\infty , +\infty)$, where $\omega$ is a pseudoconvex open set and $\psi$ is its plurisubharmonic exhaustion. He asserts that it is pseudoconvex if $u$ is plurisubharmonic.
For that he first suppose $u$ is continuous and then construct $\psi(z') + \vert \log(z_1) + u(z')\vert^{-1}$. He assert that this is an exhaustion of $\Omega$. It is easy to very that this is an exhaustion but it doesn't seem to be plurisubharmonic. How can I complement this proof?
After I post this question I realise that $-\frac{1}{x}$ is convex for $x < 0$. Is this enough for proof?