Primitive normal pairs with prescribed traces over finite fields

Let q be a positive integral power of some prime p and Fₐ㵯 be a finite field with qᵐ elements for some m N. Here we establish a sufficient condition for the existence of primitive normal pairs of the type (, f () ) in Fₐ㵯 over Fₐ with two prescribed traces, Tr₅_{ₐ㵯/Fq} () =a and Tr₅_{ₐ㵯/Fq} (f () ) =b, where f (x) Fₐ㵯 (x) is a rational function with some restrictions and a, b Fq. Furthermore, for q=5ᵏ, m 9 and rational functions with degree sum 4, we explicitly find at most 12 fields in which the desired pair may not exist.