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The inequality for (u1,u2,...,un) n-tuple

Source: IMO LongList 1982 - P28

March 18, 2011
inequalitiesinequalities unsolved

Problem Statement

Let (u1,…,un)(u_1, \ldots, u_n) be an ordered nntuple. For each k,1≤k≤nk, 1 \leq k \leq n, define vk=u1u2⋯ukkv_k=\sqrt[k]{u_1u_2 \cdots u_k}. Prove that ∑k=1nvk≤e⋅∑k=1nuk.\sum_{k=1}^n v_k \leq e \cdot \sum_{k=1}^n u_k. (ee is the base of the natural logarithm).
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