INEQUALITIES FOR THE RIEMANN-STIELTJES INTEGRAL OF S-DOMINATED INTEGRATORS WITH APPLICATIONS (II)

Assume that u, v : [a, b] -> IR are monotonic nondecreasing functions 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 vertical bar h(y) - h (x)vertical bar(2) <= [u (y) - u(x)] [v(y) - v(x)] for any x, y is an element of [a, b]. In this paper we show among other that then vertical bar integral(b)(a) f(t) g(t) dh (t)vertical bar(2) <= integral(b)(a)vertical bar f(t)vertical bar(2) du(t) integral(b)(a)vertical bar g(t)vertical bar(2) dv(t), for any continuous functions f, g : [a, b] -> C. Applications for the trapezoidal inequality 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.