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China Mathematical Olympiad 1990 problem3

Source: China Mathematical Olympiad 1990 problem3

October 18, 2013
functioninequalitiesinductionalgebra unsolvedalgebra

Problem Statement

A function f(x)f(x) defined for x≥0x\ge 0 satisfies the following conditions: i. for x,y≥0x,y\ge 0, f(x)f(y)≤x2f(y/2)+y2f(x/2)f(x)f(y)\le x^2f(y/2)+y^2f(x/2); ii. there exists a constant MM(M>0M>0), such that ∣f(x)∣≤M|f(x)|\le M when 0≤x≤10\le x\le 1. Prove that f(x)≤x2f(x)\le x^2.
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