Is there a particular word in English for a string of two inequalities which together give both a lower and an upper bound for a certain quantity? French has *encadrement* for things like this, as in

Pour tout réel

x, on a l’encadrement

This is often quite convenient — of course, one can say “we have the following inequalities…”, but the extra information is useful to have, and it helps avoiding too many repetitions.

__ Note__: The OED only lists

*encadré*as an English word, as a technical term in crystallography:

A crystal is named encadré, when it has facets which form kinds of squares around the planes of a more simple form already existing in the same species,

R. Jameson,A treatise on the external characters of minerals, 1805.

You could say that cos(x) is sandwiched by the two inequalities.

Yes, that’s a possibility, but it doesn’t really work as a noun.

Maybe you could say “we have the following bounds …” instead of “we have the following inequalities…”?

You can say: We have the two-sided inequality …

“two-sided inequality” is nice, I hadn’t thought of that.

As a further remark, although “encadrement” is quite frequent in French, I can’t think of a “named” mathematical result which refers to it (whereas there are many named inequalities).

“Sandwich” is certainly a noun, even though I’ve never seen it used in this way in print. “Capture” could lend itself to the relevant transformational grammar. The very novelty of “sandwich” (as opposed to other over-used terms, such as “normal”, in other contexts) might recommend it.

About “sandwich” (definitely a noun!), I meant that I find it difficult to imagine using it: “we have the following sandwich

a<b<c” just sounds terrible to my ear…“We have the following sandwich….” has a kind of Conway-esque flavor to it. But if not, how about “bracketing”, “embrace”, “corral”?

“bracket” or “bracketing” seems the best — I was going to mention it in another post today, after seeing it in my digital camera’s manual (as in “exposure bracket”).

So here is the post I had prepared…

A named mathematical result using “encadrement” is the calculus “théorème d’encadrement” also named “théorème des gendarmes”.