I think f(x) = -x^(-c)*log(x) is convex for any positive constant c > 0, but I don’t know how to correctly express it in CVX. I tried -x^(-c+1)*entr(x), but it didn’t work. I would appreciate your help. Thank you in advance.
As easily seen by looking at the 2nd derivative, for c = 2, it is convex for 0 < x < exp(5/6), and concave for x > exp(5/6). So your thinking is incorrect.