MathNet 2024-part-04 — 168 problems (EN–VI) Question 47

Let $a$, $b$ and $c$ be real numbers such that $0 \leq a \leq b \leq c$ and $a+b+c=1$. Prove that
$$
a b \sqrt{b-a}+b c \sqrt{c-b}+a c \sqrt{c-a}\lt \frac{1}{4}
$$