Trapezoidal Type Inequalities for Riemann-Stieltjes Integral Via Čebyšev Functional with Applications

Abstract

Some new inequalities for the functional \begin{align*} & E_{T}\left( f,u\right) \\ & :=f\left( b\right) \left( u\left( b\right) -\frac{1}{b-a}% \int_{a}^{b}u\left( t\right) dt\right) +f\left( a\right) \left( \frac{1}{b-a}% \int_{a}^{b}u\left( t\right) dt-u\left( a\right) \right) \\ & -\int_{a}^{b}f\left( t\right) du\left( t\right) , \end{align*} under various assumptions for the functions $f$ and $u$ are given. Applications for functions of selfadjoint operators and unitary operators on complex Hilbert spaces are also provided.