Ordering under assumptions

Question

I have a three-element list which contains inverses of a function u, like so:

CEs = {(u^(-1))[0.448165 u[10]+0.551835 u[30]],(u^(-1))[0.296264 u[10]+0.703736 u[30]],30}

I would now like to obtain the ranking of the elements of this list under the assumption that u is strictly increasing but struggle even to understand the output of

Ordering@Ordering@CEs

, which gives

{2, 3, 1}

without Mathematica knowing anything about u. Could someone explain to me how this ranking comes about and how I might be able to teach Mathematica that u is strictly increasing and to order elements under this assumption? In the current case the ordering should be

{1, 2, 3}

Answer 1

If $ u $ is strictly increasing, then so is $ u^{-1}. $ You don't really care about their values so a representative of strictly increasing functions will preserve the ordering. $ y=x $ is such a function so I am replacing your u, u^(-1) with Identity and the ordering is the desired:

CEs = {(u^(-1))[0.448165 u[10] + 0.551835 u[30]], (u^(-1))[
   0.296264 u[10] + 0.703736 u[30]], 30};
CEs /. Thread[{u, u^(-1)} -> Identity] // Ordering
(*{1,2,3}*)