Primitive normal pairs with prescribed traces over finite fields

Let q be a positive integral power of some prime p and Fqᵐ 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 (ε, f(ε)) in Fqᵐ over Fq with two prescribed traces, Tr_Fqᵐ/Fq(ε)=a and Tr_Fqᵐ/Fq(f(ε))=b, where f(x) ∈ Fqᵐ(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.