I am looking for some metric for distribution with support on the interval $[0,1-\epsilon]$, that will be based on the ratio of their moments. That is, if $X\sim f(x)$, $Y\sim g(y)$, I'm looking for a metric $d(f,g)$ such that $\frac{\lvert\mathbb{E}X^k-\mathbb{E}Y^k\rvert}{\mathbb{E}X^k}$ is small for all $0<k$ $\iff$ $d(f,g)$ is small.

Of course, I could just define the distance to be the sum of these ratios, but I am not sure what this means. Can two distributions be very different but have a very close moment-ratio?

I was able to show that the 1-Wasserstein distance is small if $\lvert\mathbb{E}X^k-\mathbb{E}Y^k\rvert$ is small, and vice-versa, but this is not strong enough, I want the ratio of the moments.

Is there some natural metric to look at?