Properties of the Level Sets of Some Products of Functions

We are interested about pairs $(f,g)$ of $C^2$-smooth functions $f,g:\mathbb{R}^n\longrightarrow\mathbb{R}$ with bounded ${\rm Hess}^+$ complements such that their product preserves this property as well. Recall that ${\rm Hess}^+(f)$ stands for the set of all points $p\in\mathbb{R}^n$ such that the Hessian matrix $H_p(f)$ of the $C^2$-smooth function $f\colon \mathbb{R}^n\longrightarrow\mathbb{R}$ is positive definite. In this paper we consider two pairs of real-valued functions with empty ${\rm Hess}^+$ complements whose products happen to have bounded ${\rm Hess}^+$ complements.