On the existence of pairs of primitive and normal elements over finite fields

Let ${\mathbb{F}}_{q^n}$ be a finite field with $q^n$ elements, and let $m_1$ and $m_2$ be positive integers. Given polynomials $f_1(x), f_2(x) \in $\mathbb{F}$_q[x]$with$$\text{deg}$(f_i(x)) \leq m_i$,for$i = 1, 2$, and such that the rational function $f_1(x)\mathrm{/}{f}_2(x)$ belongs to a certain set which we define, we present a sufficient condition for the existence of a primitive element $\alpha \in {\mathbb{F}}_{q^n}$, normal over ${\mathbb{F}}_q$, such that $f_1(\alpha )\mathrm{/}{f}_2(\alpha )$ is also primitive.