MathDB

9

Part of 1979 IMO Longlists

Problems(1)

Show that AM ≥ GM ≥ HM! Appeared on ILL 1979 (P9)

Source:

5/29/2011
The real numbers α1,α2,α3,…,αn\alpha_1 , \alpha_2, \alpha_3, \ldots, \alpha_n are positive. Let us denote by h=n1/α1+1/α2+⋯+1/αnh = \frac{n}{1/\alpha_1 + 1/\alpha_2 + \cdots + 1/\alpha_n} the harmonic mean, g=α1α2⋯αnng=\sqrt[n]{\alpha_1\alpha_2\cdots \alpha_n} the geometric mean, and a=α1+α2+⋯+αnna=\frac{\alpha_1+\alpha_2+\cdots + \alpha_n}{n} the arithmetic mean. Prove that h≤g≤ah \leq g \leq a, and that each of the equalities implies the other one.
inequalitiesfunctionlogarithms