Functional inequality involving reciprocals and a square root
Source: 2022 Abelkonkurransen Finale, Problem 4a
March 12, 2024
Functional inequalityFunctional Equationsalgebra
Problem Statement
Find all functions 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 .