Integral inequalities and equalities for the rearrangement of Hardy and Littlewood
Suppose \(f: [0, 1]\to \mathbb{R}\) is differentiable a.e., \(\Phi: \mathbb{R}\to [0, \infty)\) is Borel measurable, and that \(\Psi: [0, \infty)\to \mathbb{R}\) is increasing. Let \(f^*\) stand for non-increasing rearrangement of \(f\). The author proves that \[ \int^ 1_ 0 \Phi(f^* (x)) \Psi(| (f^*)'|) dx\leq \int^ 1_ 0 \Phi(f(x)) \Psi(| f'(x)|) dx\leqno{(*)} \] and, under some additional assumptions on \(f\), \(\Phi\), and \(\Psi\), he investigates the case of equality in \((*)\).
scientific article; zbMATH DE number 599911