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The a-th root of 8

Source: China Team Selection Test 2016 Test 3 Day 1 Q1

March 25, 2016
inequalitiesalgebra

Problem Statement

Let nn be an integer greater than 11, α\alpha is a real, 0<α<20<\alpha < 2, a1,…,an,c1,…,cna_1,\ldots ,a_n,c_1,\ldots ,c_n are all positive numbers. For y>0y>0, let f(y)=(∑ai≤yciai2)12+(∑ai>yciaiα)1α.f(y)=\left(\sum_{a_i\le y} c_ia_i^2\right)^{\frac{1}{2}}+\left(\sum_{a_i>y} c_ia_i^{\alpha} \right)^{\frac{1}{\alpha}}. If positive number xx satisfies x≥f(y)x\ge f(y) (for some yy), prove that f(x)≤81α⋅xf(x)\le 8^{\frac{1}{\alpha}}\cdot x.
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