On the inner ring, , you want to find critical points of . That is, the function is constant on that circle. Similarly, on the outer ring (you say "x^2+ y^2>= 16" but that can't be right. The ring is given by ) a constant. Since there is no critical point inside the ring and the function is constant on both outer and inner boundaries, there are no "critical points". The minumum value of the function is ln(4) which it attains at every point on the inner boundary and the maximum function is ln(16) which it attains at every point on the outer boundary.
In fact, you could have made us of the symmetry by putting this in polar coordinates. The function would be , which is clearly an increasing function.