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functional inequality f(xy) <= 1/2 ( (f(x) + f(y) )

Source: Norwegian Mathematical Olympiad 2011 - Abel Competition p3b

September 4, 2019
inequalitiesfunctionfunctional inequalitiesfunctionalalgebra

Problem Statement

Find all functions ff from the real numbers to the real numbers such that f(xy)≤12(f(x)+f(y))f(xy) \le \frac12 \left(f(x) + f(y) \right) for all real numbers xx and yy.
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