Question 151
a. Prove that $\frac{2a}{ab+1} \leq \sqrt{\frac{a}{b}}$ for all $a, b \in (0, \infty)$.
b. If $a, b, c \gt 0$ are such that $abc = 1$, prove that
$$
\frac{2a}{ab+1} + \frac{2b}{bc+1} + \frac{2c}{ac+1} \leq a \sqrt{c} + b \sqrt{a} + c \sqrt{b}
$$
b. If $a, b, c \gt 0$ are such that $abc = 1$, prove that
$$
\frac{2a}{ab+1} + \frac{2b}{bc+1} + \frac{2c}{ac+1} \leq a \sqrt{c} + b \sqrt{a} + c \sqrt{b}
$$