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f(x) < 12f (x - 1/2)+ 1/2 f (x +1/2)

Source: 1973 Swedish Mathematical Competition p6

March 26, 2021
functionalFunctional inequalityalgebra

Problem Statement

f(x)f(x) is a real valued function defined for x≥0x \geq 0 such that f(0)=0f(0) = 0, f(x+1)=f(x)+xf(x+1)=f(x)+\sqrt{x} for all xx, and f(x) < \frac{1}{2}f\left(x - \frac{1}{2}\right)+\frac{1}{2}f\left(x + \frac{1}{2}\right)   \text{for all}   x \geq \frac{1}{2} Show that f(12)f\left(\frac{1}{2}\right) is uniquely determined.
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