# Osgood condition

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?

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Strictly concave, or is falt okay? Because the one that suggests itself is to take the line that is the max of the derivatives at 0 of $h$ and $g$. –  Ray Yang Feb 4 '13 at 12:43
@RayYang: It is possible that $h'(0)=\infty$, e.g. if $h$ behaves like $-x\ln x$ near $0$. –  Jonas Meyer Sep 11 '13 at 6:42