Let $U \subseteq \mathbb{C}^m$ be open, and let $F: U \to \mathbb{C}$ be a holomorphic function, with real part $u$. We are given a subset $S \subseteq U$ given by finitely many real equalities and inequalities of the form:

$$ f_i(z_1,\ldots,z_m) = 0, \text{ for $i = 1,\ldots,k$} $$ and $$ g_j(z_1, \ldots, z_m) > 0, \text{ for $j = 1,\ldots,l$}, $$

where the $f_i$ and $g_j$ are real parts of corresponding holomorphic functions on $U$. Assume that $\bar{S} \subseteq U$ and that $\bar{S}$ is compact.

Consider the problem of minimizing $u$ restricted to $\bar{S}$.

Here is what I am hoping for. I am hoping to be able to prove, under some hypotheses, that

$$ \operatorname{max} \{ u(\mathbf{z}) ; \mathbf{z} \in \bar{S} \}, $$

(where $\mathbf{z} = (z_1,\ldots,z_m)$) cannot be attained at a point in $S$ (the interior of $\bar{S}$) unless $u$ is constant. In other words, I am hoping that the strong maximum principle holds in this setting (perhaps under some hopefully mild hypotheses).

Here is an idea, but I am not sure if it will work. Convert the constrained optimization problem to an unconstrained one, for instance using Lagrange multipliers, and then try to apply the strong maximum principle. I will have to think more about it.