Examples for increasing homeomorphisms related to $\varphi$-laplacian

Question

Let $\varphi$ be an odd function satisfying $\varphi(s)=\frac{s^3}{1+s^2}$ for $s \in \mathbb{R}_+:=[0,\infty).$ Then $\varphi'(s)=\frac{s^4+3s^2}{(1+s^2)^2}>0$ for $s \neq 0,$ so that $\varphi:\mathbb{R} \to \mathbb{R}$ is an odd, increasing homeomorphism.

I'd like to find examples of increasing homeomorphisms $\psi_1,\psi_2: \mathbb{R}_+\to \mathbb{R}_+$ satisfying $$ \varphi(s)\psi_1(t)\le \varphi(st) \le \varphi(s) \psi_2(t)~\hbox{for all }~s,t \in \mathbb{R}_+.\label{f1}\tag{F1}$$

1. Are there any examples of $\psi_1,\psi_2$ satisfying \eqref{f1}?

2. My question is related to generalized-laplacian problem. Is there any application or physical meaning for $\varphi(s)=\frac{s^3}{1+s^2}$? I would be grateful if you could give any comment related to differential equation.

3. Is there any example of $\psi_2$ satisfying \eqref{f1}? Here $\psi_2: \mathbb{R}_+\to \mathbb{R}_+$ is a function, not a homeomorphism.

Please let me know if you have any idea or comment for my questions. Thanks in advance.

Answer 1

I answer the mathematical part of your question.

A function $\psi_1$ satisfies F1 for all $s,t\in\Bbb R_+$ iff

$\frac{s^3}{1+s^2}\psi_1(t)\le \frac{s^3t^3}{1+s^2t^2}$ for all $s,t\in\Bbb R_+$ iff

$\frac{1}{1+s^2}\psi_1(t)\le \frac{t^3}{1+s^2t^2}$ for all $s>0$, $t\in\Bbb R_+$ iff

$\psi_1(t)\le \frac{t^3(1+s^2)}{1+s^2t^2}$ for all $s>0$, $t\in\Bbb R_+$ iff

$\psi_1(t)\le\inf_{s\in\Bbb R_+} t+\frac{t^3-t}{1+s^2t^2}$ for all $t\in\Bbb R_+$ iff

$\psi_1(t)\le t^3$ for all $t\in [0,1]$ and $\psi_1(t)\le t$ for all $t\ge 1$. In particular, a homeomorphism $\psi_1$ of $\Bbb R_+$ such that $\psi_1(t)=\min\{t^3,t\}$ for all $t\in \Bbb R_+$ satisfies these conditions.

Similarly, a function $\psi_2$ satisfies F1 for all $s,t\in\Bbb R_+$ iff

$\frac{s^3}{1+s^2}\psi_2(t)\ge \frac{s^3t^3}{1+s^2t^2}$ for all $s>0$, $t\in\Bbb R_+$ iff

$\frac{1}{1+s^2}\psi_2(t)\ge \frac{t^3}{1+s^2t^2}$ for all $s>0$, $t\in\Bbb R_+$ iff

$\psi_2(t)\ge \frac{t^3(1+s^2)}{1+s^2t^2}$ for all $s>0$, $t\in\Bbb R_+$ iff

$\psi_2(t)\ge\sup_{s\in\Bbb R_+} t+\frac{t^3-t}{1+s^2t^2}$ for all $t\in\Bbb R_+$ iff

$\psi_2(t)\ge t$ for all $t\in [0,1]$ and $\psi_2(t)\ge t^3$ for all $t\ge 1$. In particular, a homeomorphism $\psi_2$ of $\Bbb R_+$ such that $\psi_2(t)=\max\{t^3,t\}$ for all $t\in \Bbb R_+$ satisfies these conditions.