## 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.

### MSC:

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

### Keywords:

Grüss inequality; Chebyshev inequality; convex functions
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### References:

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