Inequalities for the Riemann–Stieltjes integral of $S$-dominated integrators with applications. I

Assume that $u,v:\left[ a,b\right] \rightarrow \mathbb{R}$ are monotonic nondecreasing on the interval $\left[ a,b\right] .$ We say that the complex-valued function $h:\left[ a,b\right] \rightarrow \mathbb{C}$ is S-dominated by the pair $\left( u,v\right) $ if
\begin{equation*} \left\vert h\left( y\right) -h\left( x\right) \right\vert ^{2}\leq \left[ u\left( y\right) -u\left( x\right) \right] \left[ v\left( y\right) -v\left( x\right) \right] \end{equation*}
for any $x,y\in \left[ a,b\right] .$ In this paper we show amongst other that
\begin{equation*} \left\vert \int_{a}^{b}f\left( t\right) dh\left( t\right) \right\vert ^{2}\leq \int_{a}^{b}\left\vert f\left( t\right) \right\vert du\left( t\right) \int_{a}^{b}\left\vert f\left( t\right) \right\vert dv\left( t\right) , \end{equation*}
for any continuous function $f:\left[ a,b\right] \rightarrow \mathbb{C}$. Applications for the trapezoidal and midpoint inequalities are given. New inequalities for some Čebyšev and (CBS)-type functionals are presented. Natural applications for continuous functions of selfadjoint and unitary operators on Hilbert spaces are provided as well.