Exercise 15, Appendix 0

Let $f$ be an (H) function, convex on a neighbourhood of $+\infty$, such that $f(x) \gg x$. For every $\alpha > 0$ let $x_\alpha$ be the point such that $f'(x_\alpha) = (1 + \alpha) f'(x)$.

a) If $f$ is of order $+\infty$ relative to $x$ show that

(use prop. 2 (V, p. 251) and 4 (V, p. 255), applying the Taylor formula to $\log f'(x)$). Show that $\left( f(x_\alpha) - (x_\alpha - x)f'(x_\alpha) \right)/f(x)$ tends to $(1 + \alpha)(1 - \log(1 + \alpha))$ as $x$ tends to $+\infty$.

b) If $f$ is of order $r > 1$ relative to $x$ show that

(remark that if $f_1$ is of order 0 relative to $x$ one has $f_1(kx) \sim f_1(x)$ for every constant $k$, by prop. 4 of V, p. 255). Deduce that $\left( f(x_\alpha) - (x_\alpha - x)f'(x_\alpha) \right)/f(x)$ tends to as $x$ tends to $+\infty$.

c) Suppose finally that $f$ is of order 1 relative to $x$. Putting $f(x) = x f_1(x)$ show that

(consider the inverse function of $f_1$, which is of order $+\infty$ relative to $x$). Deduce that $\left( f(x_\alpha) - (x_\alpha - x)f'(x_\alpha) \right)/f(x)$ tends to $-\infty$ as $x$ tends to $+\infty$.

Similarly let $x_\alpha'$ be the point such that $f'(x_\alpha') = (1 - \alpha)f'(x)$ (for $0 < \alpha < 1$). State the formulae analogous to the previous ones that express the principal part of $x - x_\alpha'$ and the limit of

as $x$ tends to $+\infty$.

Conclude from these results that $f$ is a regularly convex function on a neighbourhood of $+\infty$ (V, p. 260, exerc. 5).