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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