## Annals of Functional Analysis

### Ostrowski's Type Inequalities for‎ ‎Continuous Functions of Selfadjoint Operators on Hilbert Spaces‎: ‎a Survey of‎ ‎Recent Results

S‎. ‎S‎. Dragomir

#### Abstract

‎In this survey we present some recent results obtained by the author in‎ ‎extending Ostrowski inequality in various directions for continuous‎ ‎functions of selfadjoint operators defined on complex Hilbert spaces‎.

#### Article information

Source
Ann. Funct. Anal., Volume 2, Number 1 (2011), 139- 205.

Dates
First available in Project Euclid: 12 May 2014

https://projecteuclid.org/euclid.afa/1399900269

Digital Object Identifier
doi:10.15352/afa/1399900269

Mathematical Reviews number (MathSciNet)
MR2811214

Zentralblatt MATH identifier
1231.47012

Subjects
Primary: ‎47A63
Secondary: 47A99: None of the above, but in this section

#### Citation

Dragomir, S‎. ‎S‎. Ostrowski's Type Inequalities for‎ ‎Continuous Functions of Selfadjoint Operators on Hilbert Spaces‎: ‎a Survey of‎ ‎Recent Results. Ann. Funct. Anal. 2 (2011), no. 1, 139-- 205. doi:10.15352/afa/1399900269. https://projecteuclid.org/euclid.afa/1399900269

#### References

• G.A. Anastassiou, Univariate Ostrowski inequalities revisited, Monatsh. Math. 135 (2002), no. 3, 175–189.
• L. Arambašić, D. Bakić and M.S. Moslehian, A treatment of the Cauchy-Schwarz inequality in $C$*-modules, J. Math. Anal. Appl. 381 (2011), 546–556.
• L. Arambašić and R. Rajić, Ostrowski's inequality in pre-Hilbert $C$-modules, Math. Inequal. Appl. 12 (2009), no. 1, 217–226.
• P. Cerone and S.S. Dragomir, Midpoint-type rules from an inequalities point of view, Ed. G. A. Anastassiou, Handbook of Analytic-Computational Methods in Applied Mathematics, CRC Press, New York. 135–200.
• P. Cerone and S.S. Dragomir, New bounds for the three-point rule involving the Riemann–Stieltjes integrals, in Advances in Statistics Combinatorics and Related Areas, C. Gulati, et al. (Eds.), World Science Publishing, 2002, 53–62.
• P. Cerone and S.S. Dragomir, New bounds for the three-point rule involving the Riemann–Stieltjes integrals, in Advances in Statistics Combinatorics and Related Areas, C. Gulati, et al. (Eds.), World Science Publishing, 2002, 53–62.
• P. Cerone, S.S. Dragomir and J. Roumeliotis, Some Ostrowski type inequalities for n-time differentiable mappings and applications, Demonstratio Math. 32(2) (1999), 697–712.
• A. Čivljak and L. Dedić, Generalizations of Ostrowski inequality via biparametric Euler harmonic identities for measures, Banach J. Math. Anal. 4 (2010), no. 1, 170–184.
• S.S. Dragomir, Some Ostrowski's type vector inequalities for functions of selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll. 13 (2010), (to appear).
• S.S. Dragomir, Error estimates in approximating functions of selfadjoint operators in Hilbert spaces via a Montgomery's type expansion, Preprint RGMIA Res. Rep. Coll. (to appear).
• S.S. Dragomir, An Ostrowski like inequality for convex functions and applications, Revista Math. Complutense 16 (2) (2003), 373–382.
• S.S. Dragomir, On the Ostrowski's inequality for Riemann–Stieltjes integral, Korean J. Appl. Math. 7 (2000), 477–485.
• S.S. Dragomir, On the Ostrowski inequality for Riemann–Stieltjes integral $\int_{a}^{b}f\left( t\right) du\left( t\right)$ where $f$ is of Hölder type and $u$ is of bounded variation and applications, J. KSIAM 5(1) (2001), 35–45.
• S.S. Dragomir, Ostrowski's inequality for monotonous mappings and applications, J. KSIAM 3(1) (1999), 127–135.
• S.S. Dragomir, The Ostrowski's integral inequality for Lipschitzian mappings and applications, Comp. and Math. with Appl., 38 (1999), 33–37.
• S.S. Dragomir, Ostrowski type inequalities for isotonic linear functionals, J. Inequal. Pure Appl. Math. 3(5) (2002), Art. 68.
• S.S. Dragomir, Ostrowski's type inequalities for some classes of continuous functions of selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll. (to appear).
• S.S. Dragomir, Some Ostrowski's Type Vector Inequalities for Functions of Selfadjoint Operators in Hilbert Spaces, Preprint RGMIA Res. Rep. Coll. (to appear).
• S.S. Dragomir, The Ostrowski's integral inequality for Lipschitzian mappings and applications, Comp. Math. Appl. 38 (1999), 33–37.
• S.S. Dragomir, Ostrowski type inequalities for isotonic linear functionals, J. Inequal. Pure Appl. Math. 3(5) (2002), Art. 68.
• S.S. Dragomir, Inequalities of Grüss type for the Stieltjes integral and applications, Kragujevac J. Math. 26 (2004), 89–112.
• S.S. Dragomir, A generalisation of Cerone's identity and applications, Tamsui Oxf. J. Math. Sci. 23(1) (2007), 79–90.
• S.S. Dragomir, Accurate approximations for the Riemann–Stieltjes integral via theory of inequalities, J. Math. Inequal. 3(4) (2009), 663–681.
• S.S. Dragomir, Bounds for the difference between functions of selfadjoint operators in Hilbert spaces and integral means, PreprintRGMIA Res. Rep. Coll. (to appear).
• S.S. Dragomir, On the Ostrowski's inequality for mappings of bounded variation and applications, Math. Ineq. Appl. 4(1) (2001), 33-40.
• S.S. Dragomir, The Ostrowski's integral inequality for Lipschitzian mappings and applications, Comp. and Math. with Appl. 38 (1999), 33–37.
• S.S. Dragomir,Ostrowski type inequalities for isotonic linear functionals, J. Inequal. Pure Appl. Math. 3(5) (2002), Art. 68.
• S.S. Dragomir, Comparison between functions of selfadjoint operators in Hilbert spaces and integral means, Preprint RGMIA Res. Rep. Coll. (to appear).
• S.S. Dragomir, Ostrowski's type inequalities for some classes of continuous functions of selfadjoint operators in Hilbert spaces, RGMIA Res. Rep. Coll. (to appear).
• S.S. Dragomir, On the Ostrowski inequality for the Riemann–Stieltjes integral $\int_{a}^{b}f\left( t\right) du\left( t\right)$, where $f$ is of Hölder type and $u$ is of bounded variation and applications, J. KSIAM, 5 (2001), No. 1, 35–45.
• S.S. Dragomir, Inequalities for the Čebyšev functional of two functions of selfadjoint operators in Hilbert spaces, RGMIA Res. Rep. Coll. 11(e) (2008), Art. 13.
• S.S. Dragomir, Ostrowski's type inequalities for Hö lder continuous functions of selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll. (to appear).
• S.S. Dragomir, P. Cerone, J. Roumeliotis and S. Wang, A weighted version of Ostrowski inequality for mappings of Hölder type and applications in numerical analysis, Bull. Math. Soc. Sci. Math. Romanic. 42(90)(4) (1999), 301–314.
• S.S. Dragomir and Th. M. Rassias (Eds), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publisher, 2002.
• S.S. Dragomir and S. Wang, A new inequality of Ostrowski's type in $L_{1}-$norm and applications to some special means and to some numerical quadrature rules, Tamkang J. of Math. 28 (1997), 239–244.
• S.S. Dragomir and S. Wang, Applications of Ostrowski's inequality to the estimation of error bounds for some special means and some numerical quadrature rules, Appl. Math. Lett. 11 (1998), 105–109.
• S.S. Dragomir and S. Wang, A new inequality of Ostrowski's type in $L_{p}-$norm and applications to some special means and to some numerical quadrature rules, Indian J. of Math. 40(3) (1998), 245–304.
• S.S. Dragomir and I. Fedotov, An inequality of Grüss type for Riemann–Stieltjes integral and applications for special means, Tamkang J. Math. 29(4) (1998), 287–292.
• S.S. Dragomir and I. Fedotov, A Grüss type inequality for mappings of bounded variation and applications to numerical analysis, Non. Funct. Anal. Appl. 6(3) (2001), 425–437.
• A.M. Fink, Bounds on the deviation of a function from its averages, Czechoslovak Math. J. 42(117) (1992), No. 2, 298–310.
• T. Furuta, J. Mićić Hot, J. Pečarić and Y. Seo, Mond-Pečarić Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.
• G. Helmberg,Introduction to Spectral Theory in Hilbert Space, John Wiley & Sons, Inc., New York, 1969.
• Z. Liu, Refinement of an inequality of Grüss type for Riemann–Stieltjes integral, Soochow J. Math. 30(4) (2004), 483–489.
• C.A. McCarthy, $c_{p}$, Israel J. Math. 5 (1967), 249–271.
• B. Mond and J. Pečarić, Convex inequalities in Hilbert spaces, Houston J. Math. 19(1993), 405–420.
• A. Ostrowski, Über die Absolutabweichung einer differentienbaren Funktionen von ihren Integralmittelwert, Comment. Math. Hel. 10(1938), 226–227.