Under what conditions is the implication $I(x) < I(y) \implies x < y$ true?

Question

Let $\sigma(x)$ be the sum of the divisors of $x$, and denote the abundancy index of $x$ by $$I(x) = \frac{\sigma(x)}{x}.$$

My question is:

Under what conditions on $x$ and $y$ is the implication $I(x) < I(y) \implies x < y$ true?

Answer 1

One condition that I could think of is $$\frac{\sigma(x)}{\sigma(y)} < \frac{x}{y} < 1.$$

Another one is $$\left(x \mid y\right) \land \left(x \neq y\right).$$

The case of greatest interest is of course when $\gcd(x, y) = 1$.