Divergence minimization under prior inequality constraints

Motivated by problems in robust statistics we first give a simple proof of the following: Given a probability measure P and positive measures μ<ν, the γ-divergence from P of probability measures Q satisfying μ⩽Q or μ⩽Q⩽ν is minimized by an explicitly determined Q^* not depending on (the convex function) γ. Next we address γ-divergence minimization under the above inequality constraint and additional moment constraints