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Miklos Schweitzer 1967_4

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October 6, 2008
logarithmsreal analysisreal analysis unsolvedMiklos Schweitzerinequalities

Problem Statement

Let a1,a2,...,aN a_1,a_2,...,a_N be positive real numbers whose sum equals 1 1. For a natural number i i, let ni n_i denote the number of ak a_k for which 21−i≥ak≥2−i 2^{1-i} \geq a_k \geq 2^{-i} holds. Prove that ∑i=1∞ni2−i≤4+log⁡2N. \sum_{i=1}^{\infty} \sqrt{n_i2^{-i}} \leq 4+\sqrt{\log_2 N}. L. Leinder
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