4a

Part of 2022 Abelkonkurransen Finale

Problems(1)

Functional inequality involving reciprocals and a square root

Source: 2022 Abelkonkurransen Finale, Problem 4a

Find all functions

f:\mathbb R^+ \to \mathbb R^+

satisfying \begin{align*} f\left(\frac{1}{x}\right) \geq 1 - \frac{\sqrt{f(x)f\left(\frac{1}{x}\right)}}{x} \geq x^2 f(x), \end{align*} for all positive real numbers

x

Functional inequalityFunctional Equationsalgebra