Problem

Source: France JBMO TST 2017, Exam 1, Problem 1

Tags: function, inequalities



Determine all functions $f: \mathbb R \to \mathbb R$ which satisfy the inequality $$f(a^2) - f(b^2) \leq \left( f(a) + b\right)\left( a - f(b)\right),$$for all $a,b \in \mathbb R$.