MathNet 2025 — Part 01 (100 complete bilingual questions) Question 12

Let $f: \mathbb{R}^+ \to \mathbb{R}^+$ be a non-constant function. Prove that there exist $a, b \in \mathbb{R}^+$ such that
$$
f(a) + f(b) \gt 2f(\sqrt{ab}).
$$