# Does the maximum principle hold in this pluriharmonic setting?

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.