Exercise 5, § 3

Let $f$ be a convex real function defined on a neighbourhood of $+\infty$ such that $f(x) \gg x$. One says that $f$ is regularly convex on a neighbourhood of $+\infty$ if, for every convex function $g$ defined on a neighbourhood of $+\infty$ and such that $f \sim g$ one also has $f'_d \sim g'_d$.

For every number $\alpha > 0$ and $x$ sufficiently large let $k(\alpha, x)$ be the infimum of the numbers $\left( f(y) - (y-x)f'_d(y) \right)/f(x)$ as $y$ runs through the set of numbers $\geq x$ such that $f'_d(y) \leq (1+\alpha)f'_d(x)$. Similarly let $h(\alpha, x)$ be the infimum of the numbers

as $z$ runs through the set of numbers $\leq x$ such that $f'_d(z) \geq (1-\alpha)f'_d(x)$. Let $\psi(\alpha) = \limsup_{x \to +\infty} k(\alpha, x)$, $\varphi(\alpha) = \limsup_{x \to +\infty} h(\alpha, x)$. Show that for $f$ to be regularly convex on a neighbourhood of $+\infty$ it is necessary and sufficient that for all sufficiently small $\alpha$ one has $\psi(\alpha) < 1$ and $\varphi(\alpha) < 1$.