Question 29
For real numbers $a$, $b$, $c$ satisfying $0 \le a \le b \le c$ and $a + b + c = 1$, prove that the inequality
$$
ab\sqrt{b-a} + bc\sqrt{c-b} + ca\sqrt{c-a} \lt \frac{1}{4}
$$
holds.
$$
ab\sqrt{b-a} + bc\sqrt{c-b} + ca\sqrt{c-a} \lt \frac{1}{4}
$$
holds.