Primitive normal values of rational functions over finite fields

In this paper, we consider rational functions f with some minor restrictions over the finite field F-qn, where q = p(k) for some prime p and positive integer k. We establish a sufficient condition for the existence of a pair (alpha, f(alpha)) of primitive normal elements in F-qn over F-q Moreover, for q = 2(k) and rational functions f with quadratic numerators and denominators, we explicitly find that there are at most 55 finite fields F-qn in which such a pair (alpha, f(alpha)) of primitive normal elements may not exist.