I am looking for a reference -if there is any- about how to control the following expression:
$$\mathbb{E}\left[\frac{f(X)}{g(X)}\right]\cdot\frac{\mathbb{E}[f(X)]}{\mathbb{E}[g(X)]},$$$$\mathbb{E}\left[\frac{f(X)}{g(X)}\right]\cdot\frac{\mathbb{E}[g(X)]}{\mathbb{E}[f(X)]},$$
where $f$ and $g$ are strictly positive functions and the probability measure has compact support.
The particular case that I am interested on is when
- $f,g:\mathbb{R}^{n}\to\mathbb{R}$ are $$f(x)=\exp{\left(x^{\top}Ax\right)},$$ $$g(x)=\exp{\left(x^{\top}Bx\right)},$$
- The operator norm of $A-B$ is bounded.
- The measure is the uniform probability measure on the sphere.
That is, I am interested in trying to bound
$$\frac{\int_{\mathbb{S}^{n-1}}\exp\left(x^{\top}Ax\right)dx}{\int_{\mathbb{S}^{n-1}}\exp\left(x^{\top}Bx\right)dx}\cdot \int_{\mathbb{S}^{n-1}}\frac{\exp\left(x^{\top}Ax\right)dx}{\exp\left(x^{\top}Bx\right)dx}.$$$$\frac{\int_{\mathbb{S}^{n-1}}\exp\left(x^{\top}Ax\right)dx}{\int_{\mathbb{S}^{n-1}}\exp\left(x^{\top}Bx\right)dx}\cdot \int_{\mathbb{S}^{n-1}}\frac{\exp\left(x^{\top}Bx\right)dx}{\exp\left(x^{\top}Ax\right)dx}.$$
The only way that I could bound this is by writing: \begin{align*} \frac{\int_{\mathbb{S}^{n-1}}\exp\left(x^{\top}Ax\right)dx}{\int_{\mathbb{S}^{n-1}}\exp\left(x^{\top}Bx\right)dx}\cdot \int_{\mathbb{S}^{n-1}}\frac{\exp\left(x^{\top}Ax\right)dx}{\exp\left(x^{\top}Bx\right)dx} &= \frac{\int_{\mathbb{S}^{n-1}}\exp\left(x^{\top}(A-B)x\right)\exp\left(x^{\top}Bx\right)dx}{\int_{\mathbb{S}^{n-1}}\exp\left(x^{\top}Bx\right)dx}\cdot \int_{\mathbb{S}^{n-1}}\exp\left(x^{\top}(A-B)x\right)dx\\ &\leq \exp\left(2\left\|A-B\right\|_{op}\right), \end{align*}\begin{align*} \frac{\int_{\mathbb{S}^{n-1}}\exp\left(x^{\top}Ax\right)dx}{\int_{\mathbb{S}^{n-1}}\exp\left(x^{\top}Bx\right)dx}\cdot \int_{\mathbb{S}^{n-1}}\frac{\exp\left(x^{\top}Bx\right)dx}{\exp\left(x^{\top}Ax\right)dx} &= \frac{\int_{\mathbb{S}^{n-1}}\exp\left(x^{\top}(A-B)x\right)\exp\left(x^{\top}Bx\right)dx}{\int_{\mathbb{S}^{n-1}}\exp\left(x^{\top}Bx\right)dx}\cdot \int_{\mathbb{S}^{n-1}}\exp\left(x^{\top}(B-A)x\right)dx\\ &\leq \exp\left(2\left\|A-B\right\|_{op}\right), \end{align*}
where $\left\|\cdot\right\|_{op}$ is the operator norm.
The exponential that appears in the bound is a problem for me. If anyone know if something like this can have a better bound (or if it cannot be establish a better one) or have a reference to share I will be very grateful.
Edit: I messed up the orden in which the quotients are taken.