Exercise 39, § 1

Let $m$ be a convex function on $[0, +\infty[$ such that $m(0) = 0$, and $p$ a number such that $-1 < p < +\infty$. Show that the integral $\int_0^{+\infty} x^p \exp(-m'(x)) dx$ is convergent, as is the integral $\int_0^{+\infty} x^p \exp(-m(x)/x) dx$, and that

(For $k > 1$ and $A > 0$ note that $m(kx) \geq m(x) + (k - 1)x\ m'_r(x)$, and deduce the inequality

Estimate the second integral with the help of Hölder’s Inequality (III, p. 115, exerc. 3) then let $A$ tend to $+\infty$ and $k$ tend to 1.)