# ca.classical analysis and odes – Increasing concave functions bounded between linear and quadratic

Are there any explicit examples of twice-differentiable functions $$g$$, defined on the nonnegative real axis $$(0,infty)$$, such that

• $$g'(x) ge0$$
• $$g”(x)le0$$,

and such that

• For $$xge K$$ for any fixed number $$Kge1$$ (you have the freedom to choose any $$Kge1$$ here), we have the property $$|g”(x)| > C/x$$, preferably $$|g”(x)| ge C/x^{1-epsilon}$$, for some $$epsilon>0$$?

Here $$C$$ is a constant that can be any nonzero number.