Let $K$ and $K^\prime$ positive definite $n \times n$ matrices, such that for all vectors $f \ge 0$ with **nonnegative** coordinates we have

$$\sum_{i,j} K_{ij} f_i f_j \le \sum_{ij} K^\prime_{ij} f_i f_j$$

Can someone prove the following inequality?

$$ \sum_{i,j} \exp K_{ij} \le \sum_{i,j} \exp K^\prime_{ij} $$

The reason why I believe it is true is that I have a proof using Slepian-Kahane-type comparison for Wick exponentials of Gaussian random vectors with covariance $K$ and $K^\prime$, which might look like a perverse way of doing something as simple-looking as this. Ideally, I'd like to see a more straightforward proof, or maybe some general suggestions on how to use that weird positivity condition on $K^\prime - K$.

**Upd:** The "weird positivity condition" is nothing but dual to something known by the name "complete positivity".

nonnegativityhypothesis, which (despite its beingbold), I missed seeing... – Suvrit Mar 23 at 19:33