A rational function in $ \mathbb{R}[x_1, x_2] $ is called positive if $f = g/h$ with $g,h \in \mathbb{R}_{\geq 0}[x_1, x_2]$. Are there some references about the following theorem given by Poincare?
Theorem. A rational function $f$ is positive if $f\big((\mathbb{R}_{>0})^2\big) \subset \mathbb{R}_{>0}$.
As it stands, that is not true (assuming you mean by ${\bf R}_{\geq 0}[x_1, x_2]$ the set of polynomials with only nonnegative coefficients); for example, $f = 1 + (x-y)^2$ is strictly positive on the positive orthant, but cannot be so expressed. Perhaps you were thinking of the one variable result, due to Poincaré, if $f > 0$ on ${\bf R}^+\setminus \{0\}$, then there exists an integer $n$ such that $(1+x)^n f$ has no negative coefficients (so $f = ((1+x)^n f)/(1+x)^n$).
There is a general necessary and sufficient condition (in any number of variables): suppose $f$ is a real polynomial in $n$ variables, and form its Newton polytope, the convex hull of the exponents of monomials in $f$. For each face $F$ of dimension one or more, set $f_F$ to be the piece of $f$ whose monomials' exponents lie in $F$. Then $f = g/h$ with $g,h$ having no nonnegative coefficients iff for every face $F$ (of dimension at least one) of the Newton polytope (including $F$ equals the whole thing) $f_F > 0$ (as a function on the strictly positive orthant$^*$). When this occurs, the denominator can be chosen to be a sufficiently large power of anything with strictly positive coefficients at the lattice points of the Newton polytope, and no other terms.
This, together with references to Poincaré, Meissner, and Polya, who did relevant work on this type of problem, can be found in one or the other of my monographs, Positive polynomials and product type actions of compact groups, Memoirs AMS 320, or Positive Polynomials, Convex Integral Polytopes, and a Random Walk Problem SLN 1282 (1987) (at the moment, I can't locate copies of either, and I can't remember which of them it's in).
In particular, $1 + (x-y)^2$ is strictly positive on the interior of the positive orthant, but its restriction to one of the faces of the Newton polytope is $(x-y)^2$, which clearly has a zero in the interior of the positive quadrant. Therefore, this choice for $f$ fails to be such a quotient.
For the more general problem, given $P$ with nonnegative coefficients, and $f$, iff conditions for there to be $n$ such that $P^n f$ has no negative coefficients are given in Deciding eventual positivity of polynomials, Ergodic theory and dynamical systems 6 (1986) 57-79, also by me.
$^*$ When $F$ is a proper face (that is, not the whole thing), $f_F$ can effectively be written as a polynomial in fewer variables, so strict positivity is easier to verify, the lower the dimension of $F$. Positivity at the zero-dimensional faces comes for free from $f > 0$ on the positive orthant.
