harmonic analysis – Minimizing the real part of a complex analytic function on a compact totally real submanifold


Has the problem of minimizing the real part of a holomorphic/meromorphic function on a Kähler manifold restricted to a smooth compact totally real submanifold been studied before? What kind of inequalities may hold for instance? I have similar questions for the absolute value of a holomorphic function, or perhaps the log of the absolute value of a holomorphic function (defined on a Kähler manifold), also restricted to a compact totally real submanifold.