For any set of positive real numbers, the relationship $HM \le GM \le AM$ holds.

For a set of distinct positive real numbers, the strict inequality $HM < GM < AM$ holds.

If all numbers in the set are equal to a constant $k > 0$, then $HM = GM = AM = k$.

For any set of positive real numbers, $GM = \sqrt{AM \cdot HM}$ holds, regardless of the size of the set.