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f(x)+f(y) <=f(x+y) if f(x)/x is increasing for x>0

Source: 2008 Swedish Mathematical Competition p3

April 27, 2021
algebrainequalities

Problem Statement

The function f(x)f(x) has the property that f(x)x\frac{f(x)}{x} is increasing for x>0x>0. Show that f(x)+f(y)≤f(x+y),for all x,y>0 f(x)+f(y) \leq f(x+y) \qquad , \qquad \text{for all } x,y>0
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