Primitive normal values of rational functions over finite fields

In this paper, we consider rational functions [Formula: see text] with some minor restrictions over the finite field [Formula: see text] where [Formula: see text] for some prime [Formula: see text] and positive integer [Formula: see text]. We establish a sufficient condition for the existence of a pair [Formula: see text] of primitive normal elements in [Formula: see text] over [Formula: see text] Moreover, for [Formula: see text] and rational functions [Formula: see text] with quadratic numerators and denominators, we explicitly find that there are at most [Formula: see text] finite fields [Formula: see text] in which such a pair [Formula: see text] of primitive normal elements may not exist.