Inequalities for Stieltjes integrals with convex integrators and applications (Q867998)

The author considers the following functional: \[ D(f:u):= \int^b_a f(x)du(x)-(u(b)-u(a)) {1\over b-a} \int^b_a f(t)\,dt \] provided that the Stieltjes integral and Riemann integral exist. The main aim of this work is to establish sharp inequalities for the functional \(D(.,.)\) on the assumption that the integrator \(u\) in the Stieltjes integral \(\int^b_a f(x) du(x)\) is convex on \([a,b]\). Applications for the Chebyshev functional of two Lebesgue integrable functions are also given.