Exercise 16, § 4

Let $f$ be a positive convex function in $]0, +\infty[$, $a$ and $b$ two arbitrary real numbers. Now prove that the function $x^a f(x^{-b})$ is convex in $]0, +\infty[$, in the following cases: 1° $a = \frac{1}{2}(b+1)$, $|b| \geqslant 1$, 2° $x^a f(x^{-b})$ increasing, $a(b-a) \geqslant 0$, $a \geqslant \frac{1}{2}(b+1)$; 3° $x^a f(x^{-b})$ decreasing, $a(b-a) \geqslant 0$, $a \leqslant \frac{1}{2}(b+1)$. Under the same hypotheses on $f$, now prove that $e^{x/2} f(e^{-x})$ is convex (use Ex. 2 of I, p. 51). \*