I asked similar questions before (see [Examples of increasing homeomorphism from $\mathbb{R}_+$ onto $\mathbb{R}_+$ satisfying some inequalities and [Examples for increasing homeomorphisms related to $\varphi$-laplacian), but I couldn't get perfect answer for my questions. So I ask them again more precisely.
[EDIT] From Alex Ravsky's answer in [Examples of increasing homeomorphism from $\mathbb{R}_+$ onto $\mathbb{R}_+$ satisfying some inequalities, I reformulate my question as follows:
Let $\varphi$ be an increasing homeomorphism from $\mathbb{R}_+:=[0,\infty)$ onto $\mathbb{R}_+$. Set $$\overline{\varphi}(t)=\sup_{s>0} \frac{\varphi(st)}{\varphi(s)}~\hbox{and}~\underline{\varphi}(t)=\inf_{s>0} \frac{\varphi(st)}{\varphi(s)}~\hbox{for}~t \in \mathbb{R}_+.$$
Assume that $\exists$ an increasing homeomorphism $\psi_1:\mathbb{R}_+\to \mathbb{R}_+$ satisfying $$\psi_1(t)\le \underline{\varphi}(t),~\forall t \in \mathbb{R}_+\label{p1}\tag{P1}.$$ I'd like to find some examples related to the following inequality $$\overline\varphi(t) \le \psi_2(t),~\forall t \in \mathbb{R}_+\label{p2}\tag{P2}.$$
$(1)$ Are there any examples of $\varphi$ such that $\not \exists$ a homeomorphism $\psi_2: \mathbb{R}_+\to \mathbb{R}_+$ satisfying \eqref{p2}?
(2) Are there any examples of $\varphi$ such that $\not \exists$ a function $\psi_2: \mathbb{R}_+\to \mathbb{R}_+$ satisfying \eqref{p2}?
(3) Are there any examples of $\varphi$ such that $\exists$ a function (not a homeomorphism) $\psi_2: \mathbb{R}_+\to \mathbb{R}_+$ satisfying \eqref{p2}?
I don't think that the existence of an increasing homeomorphism $\psi_1$ satisfying \eqref{p1} implies the existence of a function (or a homeomorphism) $\psi_2$ satisfying \eqref{p2}.
Seeing Alex Ravsky's answer, if $\exists t_1 \in (0,1)$ and $\exists t_2>1$ such that $$\overline{\varphi}(t_1)\in (0,1)~\hbox{and}~\overline{\varphi}(t_2)\in (0,\infty),$$ then one can construct an increasing homeomorphism $\psi_2:\mathbb{R}_+\to \mathbb{R}_+.$ In this case, the existence of an increasing homeomorphism $\psi_2$ seems natural, since $\overline{\varphi}$ is nondecreasing in $\mathbb{R}_+$, $\overline{\varphi}(0)=0$ and $\overline{\varphi}(1)=1.$
I tried to find suitable examples for my questions, but I couldn't. For example, for any polynomials $\varphi$ with $\varphi(0)=0$, $\exists$ a homeomorphism $\psi_1$ satisfying \eqref{p1} and $\exists$ a homeomorphism $\psi_2$ satisfying \eqref{p2}. For $\varphi(s)=e^s-1,$ $\not\exists \psi_1$ satisfying \eqref{p1} and $\not\exists \psi_2$ satisfying \eqref{p2}.
Please let me know if you have any idea or comment for my questions. Thanks in advance.
A Russian proverb says «хорошая мысля приходит опосля» that means that a good idea comes after.
It turned out that the long awaited last part of my answer is not needed to answer your question. Indeed, assume that there exists a required homeomorphism $ψ_1$ satisfying (P1). Then for each $s’,t’>0$ we have $\varphi(s’t’)\ge\varphi(s’)\psi_1(t’)>0$. Substituting $s’t’=s$ and $1/t’=t$, we have $\varphi(s)\ge\varphi(st)\psi_1(1/t)>0$ for each $s,t>0$. That is $\varphi(st)/\varphi(s)\le 1/\psi_1(1/t)$, so $\overline{\varphi}(t)\le 1/\psi_1(1/t) $. This easily implies that the points $\overline{t}_-$ and $\overline{t}_+$ from my exists, so there exists the required homeomorphism $\psi_2$ satisfying (P2).
Moreover, that my answer is not needed. Indeed, if there exists a (necessarily increasing) homeomorphism $\psi_1$ of $\mathbb{R}_+$ such that $ \varphi(s)\psi_1(t)\le \varphi(st)$ for all $s,t \in \mathbb{R}_+$ then putting $\psi_2(0)=0$ and $\psi_2(t)=1/\psi_1(1/t)$ for each $t>0$ we obtain an increasing homeomorphism of $\mathbb{R}_+$ such that $\varphi(st)\le \varphi(s)\psi_2(t)$ for all $s,t \in \mathbb{R}_+$.