I saw in the paper "Smooth Satabilization Implies Coprime Factorization" of Eduardo Sontag the following argument: Given a smooth map $a:\mathbb{R} ^{n}\rightarrow\mathbb{R}^{+}$, let $\rho$ be any smooth function of class $\mathcal{K}_{\infty}$ such that $\lim_{\xi\rightarrow\infty}a\left( \xi\right) \rho\left( \left\Vert \xi\right\Vert \right) =+\infty$. It is claimed that $\rho$ exists. Let $\alpha_{3}^{\ast}$ be any smooth function of class $\mathcal{K}_{\infty}$ so that
$ \alpha_{3}^{\ast}\left( s\right) \leq\inf\left\{ a\left( \xi\right) \rho\left( \left\Vert \xi\right\Vert \right) :\left\Vert \xi\right\Vert =s\right\} $
for all $s\geq0$.
My main concern is on the existence of such $\alpha_{3}^{\ast }\in\mathcal{K}_{\infty}$ because a positive-definite function is not always lower-bounded by a function in the class $\mathcal{K}_{\infty}$. This is used in that paper to make the following boundedness:
$-a\left( \xi\right) \rho\left( \left\Vert \xi\right\Vert \right) \leq-\alpha_{3}^{\ast}\left( \left\Vert \xi\right\Vert \right)$
