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Inequality with semiconvex sequences

Source: Romanian IMO TST 2006, day 4, problem 4

May 19, 2006
inequalitiesfunctionintegrationcalculusderivativeinequalities unsolved

Problem Statement

Let pp, qq be two integers, q≥p≥0q\geq p\geq 0. Let n≥2n \geq 2 be an integer and a0=0,a1≥0,a2,…,an−1,an=1a_0=0, a_1 \geq 0, a_2, \ldots, a_{n-1},a_n = 1 be real numbers such that ak≤ak−1+ak+12, ∀ k=1,2,…,n−1. a_{k} \leq \frac{ a_{k-1} + a_{k+1} } 2 , \ \forall \ k=1,2,\ldots, n-1 . Prove that (p+1)∑k=1n−1akp≥(q+1)∑k=1n−1akq. (p+1) \sum_{k=1}^{n-1} a_k^p \geq (q+1) \sum_{k=1}^{n-1} a_k^q .
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