Inequalities for the Riemann–Stieltjes Integral of under the Chord Functions with Applications

Dragomir, Silvestru S.

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 53 (2014) no. 1, pp. 45-64.

Résumé

We say that the function $f\colon [a,b] \rightarrow \mathbb {R}$ is under the chord if \begin{equation*} \frac{\left( b-t\right) f(a) +\left( t-a\right) f(b) }{b-a}\ge f(t) \end{equation*} for any $t\in [a,b] $. In this paper we proved amongst other that \begin{equation*} \int _{a}^{b}u(t) df(t) \ge \frac{f(b) -f(a) }{b-a}\int _{a}^{b}u(t) dt \end{equation*} provided that $u\colon [ a,b] \rightarrow \mathbb {R}$ is monotonic nondecreasing and $f\colon [a,b] \rightarrow \mathbb {R}$ is continuous and under the chord. Some particular cases for the weighted integrals in connection with the Fejér inequalities are provided. Applications for continuous functions of selfadjoint operators on Hilbert spaces are also given.

Détail

Classification : 26D15, 47A63
Keywords: Fejér inequality; functions of bounded variation; monotonic functions; total variation; selfadjoint operators