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Inequalities for the Riemann-Stieltjes integral of under the chord functions with applications. (English) Zbl 1310.26019

Summary: We say that the function \(f:[a,b] \to \mathbb{R}\) is under the chord if
\[ \frac{( b-t) f(a) +( t-a) f(b)}{b-a}\geq f(t) \]
for any \(t\in [a,b]\).
In this paper we prove amongst other that
\[ \int _a^b u(t) df(t) \geq \frac{f(b) -f(a)}{b-a}\int _a^bu(t) dt \]
provided that \(u: [ a,b] \to \mathbb{R}\) is monotonic nondecreasing and \(f: [a,b] \to \mathbb{R}\) 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.

MSC:

26D15 Inequalities for sums, series and integrals
47A63 Linear operator inequalities
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