Can we find a counterexample to the following assertion?
Assume that $f:[0,1]\to [0,1]$ is a concave increasing diffeomorphism. Then my examples say that $$\frac{1-f(x)^2}{1-x^2}\le f'(x), x\in (0,1).$$
A counterexample to: $\frac{1-f(x)^2}{1-x^2}\le f'(x)$
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