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Arithmetic and Geometric mean - [IMO LongList 1971]

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January 1, 2011
inequalitiesinequalities proposed

Problem Statement

Let a1,a2,…,ana_1, a_2, \ldots, a_n be positive numbers, mg=(a1a2⋯an)nm_g = \sqrt[n]{(a_1a_2 \cdots a_n)} their geometric mean, and ma=(a1+a2+⋯+an)nm_a = \frac{(a_1 + a_2 + \cdots + a_n)}{n} their arithmetic mean. Prove that (1+mg)n≤(1+a1)⋯(1+an)≤(1+ma)n.(1 + m_g)^n \leq (1 + a_1) \cdots(1 + a_n) \leq (1 + m_a)^n.
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