Given a multivariate function $f: \mathbb{R}_{+} \times \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$, it is "sub-homogenous" if for all $x, y \in \mathbb{R}_{+}$ and $t \in (0, 1)$, there exists $r \in (0,1)$ such that $f(x, ty) \geq t^{r}f(x,y)$ holds.
I am curious about how to verify this condition and apply it in practice. There is one method suggested by someone that checking the subhomogenity of $f(x,y)$ is equivalent to check $$\frac{\frac{\partial }{\partial y} f(x, y)}{f(x,y)}y <1$$
I've thought about it and didn't get it. Thus, I am not sure if this method is correct. If so, could anyone help to explain it in detail please? If not, could you please give some counterexample or provide alternative method to verify this subhomogenity? Many thanks in advance!