Let $f:R_+\to R_+ R_+$ be smooth on (0,\infty), $(0,\infty)$, increasing, f(0)=0 $f(0)=0$ and lim_{x\to infty}=\infty. assume $\lim_{x\to\infty}=\infty$. Assume also that f $f$ is subadditive: $f(x+y)\le f(x)+f(y) f(x)+f(y)$ for all $x,y\ge 00$. Must f $f$ be concave? The converse is obvious.