What hypothesis could I further assume in order to say if $f_n \in \Theta(g_n)$ iff $f_n \sim \gamma g_n$ for some $\gamma \in \mathbb{R}$ Following this question, I was wondering what further condition could I pose in order to assure that $f(x) \in \Theta(g(x))$ iff there's a constant $\gamma \in \mathbb{R}$ such that $f(x) \sim \gamma g(x)$. (Please assume $x \rightarrow +\infty$).
Could some condition in the second derivative help?
Update: Please keep assuming, as the previous post the monotonicity.
 A: On top of your original requirement that both $f(x)$ and $g(x)$ must be strictly monotonic, we need a very restrictive additional hypothesis. 
(A) We might require that both $f(x)$ and $g(x)$ are non-constant polynomials. Then the condition $f(x)=\Theta(g(x))$ implies that our polynomials have the same degree. Then we can take $\gamma$ to be equal to the
ratio of the leading coefficients of the two polynomials.
(B) Here is another example (also a very restrictive hypothesis):
Theorem: Let functions $f(x)$ and $g(x)$ both have the form 
$C x^a \log^b x$ with positive $a,b,C$:
$$
f(x) = C_1 x^{a_1} \log^{b_1} x, \qquad g(x) = C_2 x^{a_2} \log^{b_2} x. \tag{$*$}
$$
Then
$$
f(x)=\Theta(g(x))\quad\Leftrightarrow\quad f(x)\sim \gamma g(x) \mbox{ for some } \gamma>0.
$$
Proof: The $\Leftarrow$ direction is trivial.
The $\Rightarrow$ direction: 
Assume that $f(x)$ and $g(x)$ have the form $(*)$, and
$$f(x)=\Theta(g(x)). \tag{1}
$$
First we establish (by contradiction) that condition $(1)$ implies
$$a_1=a_2, \qquad b_1=b_2. \tag{2}  $$
Once we have $(2)$, choose
$$
\gamma = {C_1\over C_2}.
$$
Hypotheses (A) and (B) are somewhat too restrictive. It's probably enough to require that both $f(x)$ and $g(x)$ belong to a certain "nice" subset of regularly varying functions. (Maybe someone will develop this into another answer. Also, it might turn out that the monotonicity requirement is actually unnecessary if $f(x)$ and $g(x)$ belong to a "nice" subset of regularly varying functions.)
Note: Requiring that $f(x)$ and $g(x)$ be monotonic and regularly varying is not enough; even monotonic and slowly varying is not enough! Here is a counterexample:
$$
g(x) = 3\log x, \quad f(x) = 3\log x + \log x \sin \log \log x, \quad x\ge e.
$$
