Primitive normal Values of rational functions with one prescribed norm and trace over finite fields

Let q, n, m N be such that q is a prime power and a, b F. In this article we establish a sufficient condition for the existence of a primitive normal pair (, f () ) Fₐ㵯 over F with a prescribed primitive norm a and a non-zero trace b over F of, where f (x) Fₐ㵯 (x) is a rational function of degree sum n with some minor restrictions. Furthermore, for q=7ᵏ, m 7 and rational functions with numerator and denominator being linear, we explicitly find at most 6 fields in which the desired pair may not exist.