Find a function $f\in C^2(\mathbb{R})$ such that $|f^{(j)}(s)| \leq k(1+|s|)^{p-j}$ for $j=1,2$

Question

I would like to know if there is a twice differentiable function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that:

• $f(0)=0$;
• $|f'(s)|\leq c_1(1+|s|)^{p-1}$;
• $|f''(s)|\leq c_2(1+|s|)^{p-2}$;

for all $p \in \left[1,\frac{n}{n-2}\right)$, where $n \geq 3$ is a fixed natural number.

My attempt: Let $f(s)=|s|^{p-1}s$. Then,

• $f(0)=0$;

• $f'(s)=(p-1)|s|^{p-1}$.

However, following this answer, the function $f'(s)=p|s|^{p-1}$ will only be differentiable when $p-1>1$, that is, $p>2$.

Answer 1

$f(s)=c_1s$ satisfies all the conditions.