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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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    You can enclose $\TeX$ code in double dollar signs to get displayed equations, which look nicer and are easier to read.2012-04-10
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    What you have is basically Holder's inequality, which is tight (constant functions attain equality, and nonconstant functions can come arbitrarily close), so I don't think you can get anything better. *Edit*: Unless you want to involve integrals of other powers of $g$ or things like that.2012-04-10
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    In 1D I'd try some Taylor expansion, but in 3D (and with an arbitrary volume) it can be ugly.2012-04-10
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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.