I'm trying to approximate a integral of the form:
$$\int_V{g({\bf x})f({\bf x})} \; d^3x$$
Where the functions $f(\bf{x})$ and $g(\bf{x})$ are positive functions, but where only $f(\bf{x})$ is known explicitly. We do know however that the function $g(\bf{x})$ satisfies:
$$\int_V{g({\bf x})} \; d^3x~=~G$$
For some known constant $G$ and given volume $V$.
Obviously, if $f$ is a constant then we simply have $G\cdot f$, and if it's a non-constant we know that:
$$ \min(f({\bf x}))\cdot G < \int{g({\bf x})f({\bf x})} \; d^3x < \max(f({\bf x}))\cdot G$$
In the problem at hand I know $f$ is weakly varying, and the question is whether one can achieve a better approximation / tighter bound on this integral using integral inequalities or other methods. Specifically, I'd like to express the integral in terms of $G$ and some function (or integral) of $f({\bf x})$.
For further reference - the inequality stated is actually the tightest limit as @anon hypothesized. This can simply be demonstrated by taking the function $g$ to be zero everywhere expect at $\max(f)$ or at $\min(f)$ so that the inequality becomes an equality.
Therefore, unless more information is given about the function $g$, no better solution is available.