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Suppose we have functions $g\colon [0,1]\mapsto \mathbb{R}$, which is concave, vanishes at the origin and fullfills condition $$g(xy)/xy\leq g(x)/x+g(y)/y$$ for any $x,y\in[0,1]$ and $f\colon (0,\infty)\mapsto [0,1]$ which is convex and decreasing. Define function $h\colon [0,\infty)\mapsto \mathbb{R}$ as $$h(x):=g(f(x))/f(x).$$ Can it not be concave?

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@FF: Dear FF, please read the FAQ. I think this question is not suitable for MO, you might want to try posting it a math.stackexchange.com –  Chandrasekhar Apr 15 '13 at 10:00
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