Has the problem of minimizing the real part of a holomorphic function (thus of a pluriharmonic function) on a complex manifold restricted to a real smooth (let us say compact, to simplify) submanifold with boundary been studied before? What kind of inequalities may hold for instance? 

Here is what I am hoping for. I am hoping to find a lower bound of the function I am minimizing in terms of the values of the function at the boundary. So I am hoping for some kind of weak minimum principle in this setting, under some hypotheses of course.

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 a weak minimum principle. I will have to think more about it. One issue is that the submanifold is real, and not complex, so I will have to think under what conditions my approach will work. In case such ideas ring a bell, and someone has come across an article along these lines, please inform me.