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Prove that the function is concave

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September 20, 2010
functionalgebraconcavemeanIMO Shortlist

Problem Statement

A function f:I→Rf : I \to \mathbb R, defined on an interval II, is called concave if f(θx+(1−θ)y)≥θf(x)+(1−θ)f(y)f(\theta x + (1 - \theta)y) \geq \theta f(x) + (1 - \theta)f(y) for all x,y∈Ix, y \in I and 0≤θ≤10 \leq \theta \leq 1. Assume that the functions f1,…,fnf_1, \ldots , f_n, having all nonnegative values, are concave. Prove that the function (f1f2⋯fn)1/n(f_1f_2 \cdots f_n)^{1/n} is concave.
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