For a continuous, positive function $f \in C(X)$, consider the soft-minimum $$ \min^\varepsilon_{x \in X} \left\{ f(x) \right\} := - \varepsilon \log \left ( \int_X \exp(-f(x) / \varepsilon) \mathrm{d}\mu(x)\right). $$
For the usual infimum, various identities exist such as $$ \inf f+g \geq \inf f + g, $$ $$ \sup f+g \leq \sup f + g, $$ $0\leq f \leq g$: $$ \sup f \leq \sup g, \inf f \leq \inf g, $$ $$ |\sup f - \sup g| \leq \sup |f - g| $$ $$ |\inf f - \inf g| \leq \sup |f - g| $$
I'm looking for a reference which shows these identities for the soft-minimum and maximum (in case they hold), or in case they don't hold characterize the "sharpness" in some sense.