Minimum expected value over all probability functions

Question

Find the minimum value of $E[X]$ over all probability density functions f(x) satisfying the following three constraints:

(I) $f(x) = 0 $ for $x \leq 0$

(II) $ \int_{-\infty}^{\infty} f(x) dx = 1 $

(III) $h(f) = h$

Thanks!