I aim to understand "what to look for" when addressing the following class of problems.
Assume $\mathbb{R}^n$ with orthonormal coordinates $x^i$, a smooth function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ and two real fields: the gradient $a_i(x)=\partial_i f(x)$ and the Hessian $H_{ij}(x)=\partial_i \partial_j f(x)$. We restrict ourselves to the non-negative orthant $P$ defined by $x^1\geq 0,...,x^n\geq0$. We also assume that:
$a_l(x)>0$ for $l=1,...,n$ and $x \in P$
Both $a$ and $H$ have no divergences and no discontinuities in $P$
I would like to understand which tools (maybe from convex analysis?) can be used to probe the region $R \subset P$ defined by
$$ 0 \leq H_{ij}(x) \, x^ix^j \leq a_l(x) \, x^l \quad \text{ (repeated indexes are summed)} $$
Intuition tells me that studying the eigenvalues of $H$ and comparing $a$ to its eigenvectors may be useful, but I am stuck. Are these kinds of inequalities studied in any specific field? Maybe convex analysis or optimization? In particular, $H_{ij}(x) \, x^i x^j \geq 0$ is for sure true when both eigenvalues are non-negative (they are real because $H$ is symmetric). However, we may relax this thanks to the fact that $x\in P$, but how?
I am happy even with partial answers and links to references/notes discussing the concepts and techniques I may apply (i.e. I am looking for guidance).
Note: Feel free to assume $n=2$. For $n=1$, the inequalities are $0<xa'(x)< a(x)$, which has a clear geometric interpretation in terms of the graph of the positive function $a(x)$ and its tangent passing from the origin.
