Suppose I have two normal distributions, $N(0, 1/2 + \sigma^2)$ and $N(0, 1/2 + \bar{\sigma}^2)$, with $\bar{\sigma} > \sigma$.
I want to prove that this ratio: $$\frac{\int_{0}^{(2\bar{\sigma}^2 + 1) x}f_{\bar{\sigma}^2}(t)dt}{\int_{0}^{(2\sigma^2 + 1) x}f_{\sigma^2}(t)dt},$$ is decreasing in $x$, where $f_{\bar{\sigma}^2}$ is the PDF of $x \sim N(0, 1/2 + \bar{\sigma}^2)$ and $f_{\sigma^2}$ is the PDF of $x \sim N(0, 1/2 + \sigma^2)$. I've played around with graphs and am all certain its true (not to mention it seems obvious) but when I've tried doing it directly by taking the derivative, signing the derivative is really hard since it ends up having a bunch of error functions in it. Any thoughts on how to do this? Thanks!