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

Silvestru S. Dragomir

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2014)

  • Volume: 53, Issue: 1, page 45-64
  • ISSN: 0231-9721

Abstract

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We say that the function f : [ a , b ] → ℝ is under the chord if b - t f ( a ) + t - a f ( b ) b - a ≥ f ( t ) for any t ∈ [ a , b ] . In this paper we proved amongst other that ∫ a b u ( t ) d f ( t ) ≥ f ( b ) - f ( a ) b - a ∫ a b u ( t ) d t provided that u : [ a , b ] → ℝ is monotonic nondecreasing and f : [ a , b ] → 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.

How to cite

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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 53.1 (2014): 45-64. <http://eudml.org/doc/261953>.

@article{Dragomir2014,
abstract = {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.},
author = {Dragomir, Silvestru S.},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Fejér inequality; functions of bounded variation; monotonic functions; total variation; selfadjoint operators; Fejér inequality; functions of bounded variation; monotonic functions; total variation; self-adjoint operators},
language = {eng},
number = {1},
pages = {45-64},
publisher = {Palacký University Olomouc},
title = {Inequalities for the Riemann–Stieltjes Integral of under the Chord Functions with Applications},
url = {http://eudml.org/doc/261953},
volume = {53},
year = {2014},
}

TY - JOUR
AU - Dragomir, Silvestru S.
TI - Inequalities for the Riemann–Stieltjes Integral of under the Chord Functions with Applications
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2014
PB - Palacký University Olomouc
VL - 53
IS - 1
SP - 45
EP - 64
AB - 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.
LA - eng
KW - Fejér inequality; functions of bounded variation; monotonic functions; total variation; selfadjoint operators; Fejér inequality; functions of bounded variation; monotonic functions; total variation; self-adjoint operators
UR - http://eudml.org/doc/261953
ER -

References

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