Properties of \(\omega\)-rearrangements (Q1974240)

Let \(\omega\) be a concave modulus of continuity on \(\mathbb R _+\). Consider an absolutely continuous function \(\Psi\) on \([a, b]\) with finitely many points of extremum. Let \(\psi=\Psi ^{'}\). The paper is devoted to fine analysis of the extremal problem \[ \int^b_a h(t)\psi(t) dt\to\sup: \omega(h,t)\leq\omega(t), t\in [a, b-a] \] (here \(\omega(h, t)\) is the modulus of continuity of the function \(h\)). The extremal value of the above functional turns out to be equal to \(\int^{b-a}_0 R_{\omega}(\Psi, t)\omega^{'}(t) dt\), where \(R_{\omega}(\Psi, t)\) is the so-called extremal \(\omega\)-rearrangement of \(\Psi\). Certains inequalities are proved involving \(R_{\omega}(\Psi, t)\), the nondecreasing rearrangement of \(\Psi\), and the so-called \(\Sigma\)-rearrangement of \(\Psi\) introduced by \textit{N. Korneichuk} [``Exact constants in approximation theory'', Encyclopedia of Mathematics and Applications, Vol. 38, Cambridge University Press, Cambridge, New York (1990; Zbl 0722.41001); translation from the Russian original (1987; Zbl 0643.41002)].