The concept isn't strictly applicable here. In mathematics, you can often separate one source of variation from another, i.e. bounding the process (viz. coefficients) versus bounding the input variables -- in such a situation, a "lowest upper bound" would refer to the maximum random expectation using the minimum value of coefficients. But what we have here is too complicated to present an obvious model. Unless you give us the process, or more specifically the analytic form you think the process will take, you cannot claim a distinction between the two.
No, you're making it too hard.
I have to focus on a job coming up (well, I hope so; verification pending). Whether I feel like filling in details after it finishes remains to be seen.
In the mean time, you can think about the applicability of exercize 4c, in Advanced Calculus, by Watson Fulks, wherein two increasing, continuous functions, f and g, defined in an interval I such that f(x)>= g(x) in I are considered. Of course, their inverses exist. Call them phi
This exercize is to prove that phi
You don't need to have an explicit form of f and g to for this to be true. It is always true, regardless. The (verbal) definition of the f's and g's, as well as the inequalities, would follow from physics arguments that we all would agree on. If absolutely necessary, I'm sure one could prove the inequalities for general cases, before proceeding, but I doubt I'd bother. I'd leave such tasks for somebody who believes, e.g., that all else being equal, a body with a higher drag coefficient will fall faster than one with a lower drag coefficient. I feel quite confident that I can state the converse as a fact, without providing a proof. Don't you?
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