A primitive normal pair with prescribed prenorm

For any positive integers q, n, m with q being a prime power and n 5, we establish a condition sufficient to ensure the existence of a primitive normal pair (, f () ) in Fₐ^₍ over Fₐ such that PNₐ䂞/ₐ () =a, where aₐ is prescribed. Here f=f₁/f₂ₐ䂞 (x) is a rational function subject to some minor restrictions such that deg (f₁) +deg (f₂) =m and PNₐ䂞/ₐ () =₈=₀^n-1 (j i0 j n-1{_{^}}^qʲ). Finally, we conclude that for m=3, n 6, and q=7ᵏ where k, such a pair will exist certainly for all (q, n) except possibly 10 choices at most.