Consider two functions $\psi$ and $\varphi$ defined on the interval $(0,c)$ where $c\in(0,+\infty)$ and they exhibit the following characteristics:
$\psi$ and $\varphi$ are continuous, positive, and strictly monotonically decreasing over the interval $(0,c)$.
$\lim\limits_{x\to c}\psi(x)=0$, $\lim\limits_{x\to 0}\varphi(x) =\lim\limits_{x\to 0}\psi(x)=+\infty$.
$\displaystyle\lim\limits_{x\to\infty}\frac{\psi^{-1}(kx+b)}{\psi^{-1}(x)}=0,$$\ \text{for any} \ k>1 \ \ \text{and} \ \ b\geq 0.$
$\displaystyle\limsup_{x\to 0}\big(\varphi(x)-\varphi(k x)\big)<+\infty, \ \text{for} \ k>1.$
I need some examples of the pair $(\psi,\varphi)\not=(-\log,-\log)$ such that$$\psi^{-1}(s\varphi(xy))≍\psi^{-1}(s\varphi(x))\psi^{-1}(s\varphi(y))\;\;\text{for}\;\; x,y\in(o,c) \;\;\text{and}\;\; s\geq 0.$$