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.