INEQUALITIES FOR THE RIEMANN-STIELTJES INTEGRAL OF S-DOMINATED INTEGRATORS WITH APPLICATIONS. I

Assume that u,v : [a,b] -> R are monotonic nondecreasing on the interval [a, b]. We say that the complex-valued function h : [a, b] -> C is S-dominated by the pair (u, v) if (GRAPHIC)for any x,y epsilon [a, b]. In this paper we show amongst other that (GRAPHIC) for any continuous function f : [a, b] -> C. Applications for the trapezoidal and midpoint inequalities are given. New inequalities for some Cebysev and (CBS)-type functionals are presented. Natural applications for continuous functions of selfadjoint and unitary operators on Hilbert spaces are provided as well.