For two distinct positive numbers, which relation between the arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM) is always true?

For distinct positive a and $b,(\surd a-\surd b{)}^2$ > 0 gives $(a+b)/2$ > √$(ab)$, so AM > GM; and G$M^2$ = AM × HM forces GM > HM as well. Hence AM > GM > HM. (For $a=1$ and $b=4$: AM = 2.5, GM = 2 and HM = 1.6. Equality of all three happens only when the two numbers are equal.)