Distributing arbitrary values over a given range and target mean

Hey folks,

Forgive my lack of math nomenclature and probably ham-fisted notation/explanation; I'm very very rusty in this department. First post here, and I'm liking the LaTeX.

**Larger problem:**

I need possible approaches to re-distribute a set of positive real numbers such that , .

I need the re-distributed set to follow:

- the same ordering
- to be in the interval for a given value where
- the mean of is a given value where

**Sub-problem**

In fact, I would also be happy if there's an approach which satisfies the case where . My solution in this case was to divide all the elements of by the max value giving (to get between 0 and 1) and to apply some power such that:

But, I have no idea how to solve for given and .

Any help for either suggesting an approach to creating from or solving for would leave me fawning in gratitude.