We consider perturbations of a locally defined analytic function which has a critical point which is also a fixed point. The second derivative at the fixed point determines an inequality for the positions of the critical point, critical value and fixed point of the perturbed analytic function relative to some reference point. We apply this to the case of critical points of polynomials where the reference point is another fixed point We also use the topological description of polynomials all of whose critical points are fixed, to examine some inequalities relating the positions of critical points and critical values for polynomials which depend on the branching of the polynomial.