For all strictly monotonic increasing function $f:[0, \infty) \rightarrow [0,\infty)$ such that $f(0)=0$ and $lim_{x\rightarrow \infty} f(x)=\infty$, I would like to know if there always exists another strictly monotonic increasing function $f_2:[0, \infty) \rightarrow [0,\infty)$ with the same properties as $f$ such that $f(x_1)+f(x_2)\geq f_2(x_1+x_2)$ for all $x_1$, $x_2\geq0$.
I try to find a funtion from $f$, but I couldn't do it.
Thank you.
Let $f_2(x):=f(\frac{x}{2})$. For $x_1,x_2 \ge 0$, we can assume that $x_2 \le x_1$. Then we have $\frac{x_1+x_2}{2} \le x_1$, hence
$f_2(x_1+x_2)=f(\frac{x_1+x_2}{2}) \le f(x_1) \le f(x_1)+f(x_2)$
Fred