Doi: http://dx.doi.org/10.15446/recolma.v49n1.54165

¹Victoria University, Melbourne, Australia. Email: sever.dragomir@vu.edu.au

Some Grüss type inequalities for the Riemann-Stieltjes integral of continuous complex valued integrands defined on the complex unit circle C( 0,1) and various subclasses of integrators are given. Natural applications for functions of unitary operators in Hilbert spaces are provided.

Key words: Grüss type inequalities, Riemann-Stieltjes integral inequalities, Unitary operators in Hilbert spaces, Spectral theory, Quadrature rules.

Se proporcionan algunas desigualdades tipo Grüss para la integral de Riemann-Stieltjes de integrandos de valores continuos complejos definidos sobre el circulo unitario complejo C(0,1) y varias subclases de integradores son dados. Aplicaciones naturales para funciones de operadores unitarios en espacios de Hilbert son proporcionadas.

Palabras clave: Desigualdades de tipo Grüss, desigualdades integrales de Riemann-Stieltjes, Operadores unitarios en espacios de Hilbert, teoria espectral, reglas de cuadratura.

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References

[1] M. W. Alomari, `A Companion of Ostrowski's Inequality for the Riemann-Stieltjes Integral \int_a^bf(t)\,du(t), where f is of Bounded Variation and u is of r-H-Hölder Type and Applications´, Appl. Math. Comput. 219, 9 (2013), 4792-4799.         [ Links ]

[2] G. A. Anastassiou, `A New Expansion Formula´, Cubo Mat. Educ. 5, 1 (2003), 25-31.         [ Links ]

[3] S. S. Dragomir, `On the Ostrowski's Inequality for Riemann-Stieltjes Integral and Applications´, Korean J. Comput. & Appl. Math 7, 3 (2000), 611-627.         [ Links ]

[4] S. S. Dragomir, `Sharp Bounds of \vCeby\vsev Functional for Stieltjes Integrals and Applications´, Bull. Austral. Math. Soc. 67, 2 (2003), 257-266.         [ Links ]

[5] S. S. Dragomir, `Approximating the Riemann-Stieltjes Integral by a Trapezoidal Quadrature Rule With Applications´, Mathematical and Computer Modelling 54, (2011), 243-260.         [ Links ]

[6] S. S. Dragomir, `Inequalities of Lipschitz Type for Power Series in Banach Algebras´, RGMIA Research Report Collection 15, 77 (2012), 17 pp.. Preprint, Online http://rgmia.org/v15.php         [ Links ]

[7] G. Helmberg, Introduction to Spectral Theory in Hilbert Space, John Wiley, New York, USA, 1969.         [ Links ]

[8] Z. Liu, `Refinement of an Inequality of Grüss Type for Riemann-Stieltjes Integral´, Soochow J. Math. 30, 4 (2004), 483-489.         [ Links ]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

Doi: http://dx.doi.org/ @ARTICLE{RCMv49n1a04,
    AUTHOR  = {Dragomir, Silvestru Sever},
    TITLE   = {{Grüss Type Inequalities for Complex Functions Defined on Unit Circle with Applications for Unitary Operators in Hilbert Spaces}},
    JOURNAL = {Revista Colombiana de Matemáticas},
    YEAR    = {2015},
    volume  = {49},
    number  = {1},
    pages   = {77--94}
}