Assumptions to ensure that ratio of a function is less than ratio of first derivative

Question

I would like an increasing function $\phi: D \to \mathbb{R}_{++}$, where $D \subseteq \mathbb{R}$ is arbitrary for now, to have the property that for any $a, b \in D$ with $a < b$,

$$\frac{ \phi'(a)}{ \phi(a)} \leq \frac{ \phi'(b)}{\phi(b)} $$

For example, this occurs when $\phi$ is the exponential function. Are there any minimum conditions I can place on $\phi$ to ensure that this occurs? For example, regarding its concavity or convexity? Curvature restrictions on their own do not seem sufficient but I am not sure what else to include.

Totally fine to put restrictions on $D$ or on anything else about the function.

Answer 1

$\frac{ \phi'(a)}{ \phi(a)} \leq \frac{ \phi'(b)}{\phi(b)} \implies (\ln(\phi(a)))' \le (\ln(\phi(b)))' \implies (\ln(\phi(a)))'' \ge 0 $.

So $\ln(\phi(x))$ is convex.

Note that this requires that $\phi(x) > 0$.