Primitive Normal Values of Rational Functions over Finite Fields

In this paper, we consider rational functions $f$ with some minor restrictions over the finite field ${\mathbb{F}}_{q^n},$ where $q=p^k$ for some prime $p$ and positive integer $k$. We establish a sufficient condition for the existence of a pair $(\alpha ,f(\alpha ))$ of primitive normal elements in ${\mathbb{F}}_{q^n}$ over ${\mathbb{F}}_q\mathrm{.}$ Moreover, for $q=2^k$ and rational functions $f$ with quadratic numerators and denominators, we explicitly find that there are at most $55$ finite fields ${\mathbb{F}}_{q^n}$ in which such a pair $(\alpha ,f(\alpha ))$ of primitive normal elements may not exist.