On a special type of permutation rational functions

Let p be a prime and n be a positive integer. We consider rational functions f b ( X ) = X + 1 / ( X p - X + b ) over F p n with Tr ( b ) ≠ 0 . In Hou and Sze (Finite Fields Appl 68, Paper No. 10175, 2020), it is shown that f b ( X ) is not a permutation for p > 3 and n ≥ 5 , while it is for p = 2 , 3 and n ≥ 1 . It is conjectured that f b ( X ) is also not a permutation for p > 3 and n = 3 , 4 , which was recently proved sufficiently large primes in Bartoli and Hou (Finite Fields Appl 76, Paper No. 101904, 2021). In this note, we give a new proof for the fact that f b ( X ) is not a permutation for p > 3 and n ≥ 5 . With this proof, we also show the existence of many elements b ∈ F p n for which f b ( X ) is not a permutation for n = 3 , 4 .