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Recently I came up with a type of variational problem in stochastic process. It can be stated in the following way: Given $a$ and $b$ positive, and an increasing function $f$ on $(0,1)$ (may be not strictly, but $f$ is possibly unbounded), which satisfies the following equation:

$$ \int_{0}^{1}H_{1}(f(\alpha))d\alpha=a ,$$

$$ \int_{0}^{1}H_{2}(f(\alpha))d\alpha=b .$$

where $H_1$ and $H_2$ are positive and increasing function. In practice, we often need to consider the minimal of the functional $$ \int_{0}^{1}H(f(\alpha))d\alpha ,$$ where H is a function (may be not positive or increasing).

I want to know a general necessary condition for the minimal of the functional. Any comments will be appreciated.

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You question is too vague. If anything is allowed for $H$, of course, then the minimum is $-\infty$: take $$ H_n = -n \mbox{ on } f((0,1)) \mbox{ and } 0 \mbox{ otherwise.} $$ If you decide that you just want to avoid that, then you can that $H$ is bounded below, and then on whichever space you decide $H$ should be in, you can look for a minimising sequence...saying more than that in full generality is probably useless since you must have a space (or a set) in mind.

It is of course not a necessary condition: for a well tuned set of functions (depending on $f$) you can lift the lower bound requirement.

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