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.



variational problem related to an integral
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. 


