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The inequality for n numbers - true for all 0 ≤ α ≤ 1.

Source: IMO LongList 1982 - P51

May 16, 2011
inequalitiesinequalities proposed

Problem Statement

Let n numbers x1,x2,…,xnx_1, x_2, \ldots, x_n be chosen in such a way that 1≥x1≥x2≥⋯≥xn≥01 \geq x_1 \geq x_2 \geq \cdots \geq x_n \geq 0. Prove that (1+x1+x2+⋯+xn)α≤1+x1α+2α−1x2α+⋯+nα−1xnα(1 + x_1 + x_2 + \cdots + x_n)^\alpha \leq 1 + x_1^\alpha+ 2^{\alpha-1}x_2^\alpha+ \cdots+ n^{\alpha-1}x_n^\alpha if 0≤α≤10 \leq \alpha \leq 1.
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