Let $h$ and $g$ be continuous, non-decreasing and concave functions in the interval $[0,\infty)$ with $h(0)=g(0)=0$ and $h(x)>0$ and $g(x)>0$ for $x>0$ such that both satisfy the Osgood condition $$\int_{0+}\frac{dx}{f(x)}=\infty.$$
Does there exist a concave function $F$ such that $F(x)\geq h(x)$ and $F(x)\geq g(x)$ for all $x$, and satisfies the Osgood condition?
