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 $textrm{deg}(f_i(x)) leq m_i$, for $i = 1, 2$, and such that the rational function $f_1(x)/f_2(x)$ 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 $f_1(alpha)/f_2(alpha)$ is also primitive.