I am for example(s) of an invertible Convex or concave function $\phi: [0,\infty)\to [0, \infty)$ such that $\phi(0)=0$ and there exists $\theta>0$ and for all $s\leq t$ we have

\begin{align}\label{EqI}\tag{I} \phi\big(\theta \frac{s}{t}\big) \leq \frac{\phi(s)}{\phi(t)} \qquad\text{or equaly} \qquad \theta \leq \phi^{-1}\big(\frac{s}{t}\big)\frac{\phi^{-1}(t)}{\phi^{-1}(s)} \end{align}

The most simple class consists of polynomial functions of the form $\phi(t)= ct^p$ with $c>0$ and $p>0$.

**Question:** Are there other possible non-polynomial examples satisfying $\eqref{EqI}$?

As an attempt with $\phi(t)= e^{t^\alpha}-1$, I wonder if there is a constant $c>0$ such that

$$ \ln(t+1)\ln\left(\frac{s}{t}+1\right)\geq c\ln(s+1),\qquad \text{for all $0\leq s\leq t$}.$$