A 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 function $f$ is positive if $f((\mathbb{R}_{>0})^2) \subset \mathbb{R}_{>0}$.

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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 $f\big((\mathbb{R}_{>0})^2\big) \subset \mathbb{R}_{>0}$.