I also checked the concavity, and found it to be neither convex nor concave, even when restricted to 0 < x < 1, 0 < y < 1.
For instance, at x=0.8, y=0.2, the Hessian has eigenvalues of -0.16 and 246.06. Therefore, it can’t be used in CVX. Your only hope of using it in CVX is if the context of its use allows a reformulation without explicitly forming the function; but I doubt that is the case.