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Inequality with sequence of non negative real numbers

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December 31, 2011
inequalitieslogarithmsinequalities unsolved

Problem Statement

Let <an><a_n> be a sequence of non-negative real numbers such that am+n≤am+ana_{m+n} \le a_m +a_n for all m,n∈Nm,n \in \mathbb{N}. Prove that ∑k=1Nakk2≥aN4Nln⁡N\sum_{k=1}^{N} \frac{a_k}{k^2}\ge \frac{a_N}{4N}\ln N for any N∈NN \in \mathbb{N}, where ln⁡\ln denotes the natural logarithm.
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