Primitive pairs of rational functions with prescribed traces over finite fields

Let $q$ be a positive integral power of some prime $p$ and ${\mathbb{F}}_{q^m}$ be a finite field with $q^m$ elements for some $m\in \mathbb{N}$. Here we establish a sufficient condition for the existence of a non-zero element $\epsilon \in {\mathbb{F}}_{q^m}$, such that $(f(\epsilon ),g(\epsilon ))$ is a primitive pair in ${\mathbb{F}}_{q^m}$ with two prescribed traces, ${\text{Tr}}_{{\mathbb{F}}_{q^m}\mathrm{/}{\mathbb{F}}_q}(\epsilon )=a$ and ${\text{Tr}}_{{\mathbb{F}}_{q^m}\mathrm{/}{\mathbb{F}}_q}({\epsilon }^{-1})=b$, where $f(x),g(x)\in {\mathbb{F}}_{q^m}(x)$ are rational functions with some restrictions and $a,b\in {\mathbb{F}}_q$. Also, we show that there exists an element $\epsilon \in {\mathbb{F}}_{q^m}$ satisfying our desired properties in all but finitely many fields ${\mathbb{F}}_{q^m}$ over ${\mathbb{F}}_q$. We also calculate possible exceptional pairs explicitly for $m\ge 9$, when degree sums of both the rational functions are taken to be 3.