I'm troubling with an Exercise that comes from the book $\mathit{Holomorphic~Functions~in~Several~Complex~Variables~}$by R.Michael Range.
Exercise 2.6 Define$~r(z)$ for$~z\in \mathbb{C}^2~$by$$r(z)=Re(z_2)+|z_1|^8+\frac{15}{7}|z_1|^2Re(z_1^6)$$ and set$~D=\left \{ z \in\mathbb{C}^2:r(z)<0\right \} $.
(i)Show that $D$ is Levi pseudoconvex.
(ii)Find all points $p\in bD$ at which $D$ is strictly Levi pseudoconvex.
My attempt:I want to regrad $r(z)$ directly as the defining function for $D$ by the definition of $D$,then I can caculate the $\textbf{Levi form}$,that is $$L_p(r;t)=\sum_{i,j}\frac{\partial^2r}{\partial z_i \partial \bar{z_j}}t_i\bar{t_j}$$ and show that $L_p(r;t)\ge0,\forall z \in D,t\in T_p^{\mathbb{C}}(bD)$,so that $D$ is Levi pseudoconvex. However,notice that $Re(z)$ is not a complex differentiable function,I cannot directly regrad $r(z)$ as the defining function for $D$,I need to find the other defining function,but I have no idea.Any help is appreciated!
It doesn't matter that $\operatorname{Re} z$ (or the other terms in $r(z)$) is not complex differentiable. The derivatives are Wirtinger derivatives which are well-defined. If you write $$ r(z) = \frac12(z_2+\bar z_2) + z_1^4 \bar z_1^4 + \frac{15}{14} z_1 \bar z_1 (z_1^6 + \bar z_1^6), $$ you can do first year calculus differentiating, pretending $z_1$, $\bar z_1$, $z_2$ and $\bar z_2$ are "independent" variables.