Let $a\in \mathbb{C}$ be a fixed parameter. We condider the domain in $\mathbb{C}^3$ : $$S = \{ (z_1, z_2, z_3) \in \mathbb{C}^3 , \mathrm{Re} (z_i) \leq \frac 12 \}$$
I would like to prove that the minimum of the function $$f(z_1,z_2, z_3) = \mathrm{max} (|z_1 z_2 |, |z_2 z_3| , |z_1 z_3| )$$ on $S$, subject to the constraint $$z_1+z_2+z_3 -1- a\cdot z_1\cdot z_2\cdot z_3 = 0$$ is attained when $z_1 = z_2 = z_3$.
I feel that I'm perhaps missing an obvious simplification to reduce the problem. I thought about using Lagrange multipliers, but it gets pretty messy.
So, any help will be greatly appreciated !