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ЖУРНАЛЫ // Проблемы анализа — Issues of Analysis // Архив

Пробл. анал. Issues Anal., 2015, том 4(22), выпуск 1, страницы 11–37 (Mi pa186)

Inequalities for the Riemann–Stieltjes integral of $S$-dominated integrators with applications. I

S. S. Dragomir

Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia

Аннотация: Assume that $u,v:\left[ a,b\right] \rightarrow \mathbb{R}$ are monotonic nondecreasing on the interval $\left[ a,b\right] .$ We say that the complex-valued function $h:\left[ a,b\right] \rightarrow \mathbb{C}$ is S-dominated by the pair $\left( u,v\right) $ if
\begin{equation*} \left\vert h\left( y\right) -h\left( x\right) \right\vert ^{2}\leq \left[ u\left( y\right) -u\left( x\right) \right] \left[ v\left( y\right) -v\left( x\right) \right] \end{equation*}
for any $x,y\in \left[ a,b\right] .$ In this paper we show amongst other that
\begin{equation*} \left\vert \int_{a}^{b}f\left( t\right) dh\left( t\right) \right\vert ^{2}\leq \int_{a}^{b}\left\vert f\left( t\right) \right\vert du\left( t\right) \int_{a}^{b}\left\vert f\left( t\right) \right\vert dv\left( t\right) , \end{equation*}
for any continuous function $f:\left[ a,b\right] \rightarrow \mathbb{C}$. Applications for the trapezoidal and midpoint inequalities are given. New inequalities for some Čebyšev and (CBS)-type functionals are presented. Natural applications for continuous functions of selfadjoint and unitary operators on Hilbert spaces are provided as well.

Ключевые слова: Riemann–Stieltjes integral, functions of bounded variation, cumulative variation, selfadjoint operators, unitary operators, trapezoid and midpoint inequalities, Čebyšev and (CBS)-type functionals.

УДК: 517.51, 517.44

MSC: 26D15, 47A63

Поступила в редакцию: 14.04.2015
Исправленный вариант: 18.06.2015

Язык публикации: английский

DOI: 10.15393/j3.art.2015.2809



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