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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 $f({\bf x})$.

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    @anon - Nicely stated, I didn't notice this was actually Hölder's inequality with p = 1, q = infinity. But I was thinking that the fact that the functions are positive would help tighten that a bit. Also what would you suggest using as the value itself? should an average over f be used?2012-04-10

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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.