Inequalities for Stieltjes integrals with convex integrators and applications. (English) Zbl 1116.26004

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.


26A42 Integrals of Riemann, Stieltjes and Lebesgue type
26D15 Inequalities for sums, series and integrals
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