Exercise 16, § 4

Let $f$ be a positive convex function on $]0, +\infty[$, and $a, b$ two arbitrary real numbers. Show that the function $x^a f(x^{-b})$ is convex on $]0, +\infty[$ in the following cases:

1 $a = \frac{1}{2}(b + 1), \ |b| \geq 1;$

2 $x^a f(x^{-b})$ is increasing, $a(b - a) \geq 0,\ a \geq \frac{1}{2}(b + 1);$

3 $x^a f(x^{-b})$ is decreasing, $a(b - a) \geq 0,\ a \leq \frac{1}{2}(b + 1).$

Under the same hypotheses on $f$ show that $e^{x/2} f(e^{-x})$ is convex (use exerc. 2 of I, p. 45).