Finding a Sup of a Functional?

[cris(c)]

Hi guys,

I need some help please! Consider the following expression:

\left[1-\int_{x}^{1}F(\rho(\xi))f(\xi)d\xi\right]^{n-1}

where F:[0,1]\rightarrow [0,1] is a continuously differentiable function with F'=f, x∈[0,1], and n>2. Suppose that \rho belongs to the set of continuous and nondecreasing functions defined on [0,1]. Let C denote this set and endow it with the sup norm. I want to find a function \rho \in C such that (with x<1 fixed):

\left[1-\int_{x}^{1}F(\rho^*(\xi))f(\xi)d\xi\right]^{n-1}\geq \left[1-\int_{x}^{1}F(\rho(\xi))f(\xi)d\xi\right]^{n-1}

for all \rho \in C. Does this make any sense at all? if so, how can be sure I can find this function?

Thank you so much for your help! I truly need it!

 

[haruspex]

Consider the following expression:
\left[1-\int_{x}^{1}F(\rho(\xi))f(\xi)d\xi\right]^{n-1}
where F:[0,1]\rightarrow [0,1] is a continuously differentiable function with F'=f, x∈[0,1], and n>2. Suppose that \rho belongs to the set of continuous and nondecreasing functions defined on [0,1]. Let C denote this set and endow it with the sup norm. I want to find a function \rho \in C such that (with x<1 fixed):

\left[1-\int_{x}^{1}F(\rho^*(\xi))f(\xi)d\xi\right]^{n-1}\geq \left[1-\int_{x}^{1}F(\rho(\xi))f(\xi)d\xi\right]^{n-1}

for all \rho \in C.

Looks like a calculus of variations problem to me, and I have only a passing acquaintance with that. But first, I don't understand the role of n here. I think the integral is never more than 1. If so, it simplifies to
\int_{x}^{1}F(\rho^*(\xi))f(\xi)d\xi \leq \int_{x}^{1}F(\rho(\xi))f(\xi)d\xi
Anyway, I'm going to assume that.
Normally for calc of var one would consider \rho(\xi) = \rho^*(\xi) + δh(\xi), small δ > 0, but I don't see how to incorporate the non-decreasing aspect.
Another approach I tried was integration by parts:
\int_{x}^{1}F(\rho(\xi))f(\xi)d\xi = [F(\rho(\xi))F(\xi)]_{x}^{1} - \int_{x}^{1}F(\xi)F'(\rho(\xi))d\rho(\xi)
At least here we might be able to use d\rho(\xi) ≥ 0, but I'm just as stuck.
Just posting this in case it gives you a useful idea.