Extremal values of a certain class of nonlinear functionals under moment constraints (Q1902774)

``In the present article, we solve a problem of search for extrema of the nonlinear functional \[ J(\sigma)=\varphi\Biggl(\int^b_a \omega_1(t)d\sigma(t),\dots,\int^b_a \omega_r(t)d\sigma(t)\Biggr), \] where \(\varphi\), \(\omega_1,\dots,\omega_r\) are given continuous functions. Extremum is sought in the class of all mass distributions on the segment \([a,b]\), satisfying the constraints: \(c=\int^b_a u(t)d\sigma(t)\), where \(c\) is a given vector from \(\mathbb{R}^{n+1}\), \(u(t)\) is a given continuous vector-function. This problem generalizes the classical extremal problem connected with the moment problem where \(J(\sigma)=\int^b_a \omega(t)d\sigma(t)\)''.