Answer: E

Small Hint:

Recognize the arithmetic, geometric, and harmonic means of $a$ and $b$

Big Hint:

For unequal positive numbers, the mean inequalities are strict

Solution:

The arithmetic-geometric mean inequality gives $$\frac{a+b}{2}\gt\sqrt{ab}$$ for unequal positive $a,b.$ Applying the same inequality to $\frac{1}{a}$ and $\frac{1}{b}$ gives $$\sqrt{ab}\gt\frac{2ab}{a+b}.$$ Therefore the decreasing order is arithmetic mean, geometric mean, harmonic mean.

Thus, the correct answer is E.