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Inequality in Functional Equation

Source: Germany TST 2011 P3

April 12, 2020
algebrafunctionfunctional equation

Problem Statement

We call a function f:Q+→Q+f: \mathbb{Q}^+ \to \mathbb{Q}^+ good if for all x,y∈Q+x,y \in \mathbb{Q}^+ we have: f(x)+f(y)≥4f(x+y).f(x)+f(y)\geq 4f(x+y). a) Prove that for all good functions f:Q+→Q+f: \mathbb{Q}^+ \to \mathbb{Q}^+ and x,y,z∈Q+x,y,z \in \mathbb{Q}^+ f(x)+f(y)+f(z)≥8f(x+y+z)f(x)+f(y)+f(z) \geq 8f(x+y+z) b) Does there exists a good functions f:Q+→Q+f: \mathbb{Q}^+ \to \mathbb{Q}^+ and x,y,z∈Q+x,y,z \in \mathbb{Q}^+ such that f(x)+f(y)+f(z)<9f(x+y+z)?f(x)+f(y)+f(z) < 9f(x+y+z) ?
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