Given bounded functions $f, g: \mathbb{R}\to \mathbb{R}_+$, assume that $F(x) = \int_{0}^s f(x) \,dx $ and $G(x) = \int_{0}^s g(x) \,dx$ can be computed for any constant $s \ge 0$. (For example, take $f(x)$ and/or $g(x)$ to be the density function of a Normal or Gamma distribution.) Is it possible to obtain a tractable upper bound of $$\int_0^s \frac{f(x)}{g(x)} \,dx $$ using $F(x)$ and $G(x)$? What if in addition $f$ and $g$ are both non-decreasing and convex on $[0, \infty)$? Finally, assume that $h(x)$ satisfies the same conditions as $f$ and $g$, can we upper bound the integral $$\int_0^s h(x)\, \frac{f(x)}{g(x)} \,dx $$ using $F(x), G(x)$ and $H(x) = \int_0^s h(x)\,dx$?
Upper bound of integral of quotient/product of functions
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integration
inequality
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0Well, we can certainly bound the integral by $\int_0^s\frac{F'(x)}{G'(x)}dx$ ... – 2017-02-24
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0The upper bound should be tractable to compute using $F(x)$ and $G(X)$. – 2017-02-24