Geodesic
Inequalities for the Riemann–Stieltjes Integral of under the Chord Functions with Applications
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 53 (2014) no. 1, pp. 45-64 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 26D15, 47A63
Keywords: Fejér inequality; functions of bounded variation; monotonic functions; total variation; selfadjoint operators 
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     title = {Inequalities for the {Riemann{\textendash}Stieltjes} {Integral} of under the {Chord} {Functions} with {Applications}},
     journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
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Dragomir, Silvestru S. Inequalities for the Riemann–Stieltjes Integral of under the Chord Functions with Applications. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 53 (2014) no. 1, pp. 45-64. http://geodesic.mathdoc.fr/item/AUPO_2014_53_1_a3/