IMO 1968 SL 12

Origin: POL

Problem

If $a$ and $b$ are arbitrary positive real numbers and $m$ an integer, prove that

$$\left(1 + \frac{a}{b}\right)^m + \left(1 + \frac{b}{a}\right)^m \geq 2^{m+1}.$$

Solution

The given inequality is equivalent to

$$\frac{(a + b)^m}{b^m} + \frac{(a + b)^m}{a^m} \geq 2^{m+1},$$

which can be rewritten as

$$\left(\frac{1}{a^m} + \frac{1}{b^m}\right) \geq \frac{2^{m+1}}{(a + b)^m}.$$

Since $f(x) = \frac{1}{x^m}$ is a convex function for every $m \in \mathbb{Z}$, the last inequality immediately follows from Jensen’s inequality

$$\frac{f(a) + f(b)}{2} \geq f\left(\frac{a + b}{2}\right).$$