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Trapezoid type inequalities for complex functions defined on the unit circle with applications for unitary operators in Hilbert spaces

  • Sever S. Dragomir EMAIL logo

Abstract

Some trapezoid type inequalities for the Riemannā€“Stieltjes integral of continuous complex-valued integrands defined on the complex unit circle š’ž(0,1) and various subclasses of integrators of bounded variation are given. Natural applications for functions of unitary operators in Hilbert spaces are provided.

References

1 M. W. Alomari, Some GrĆ¼ss type inequalities for Riemannā€“Stieltjes integral and applications, Acta Math. Univ. Comenian. (N.S.) 81 (2012), 2, 211ā€“220. Search in Google Scholar

2 M. W. Alomari, A companion of Ostrowski's inequality for the Riemannā€“Stieltjes integral āˆ«abf(t)du(t), where f is of bounded variation and u is of r-H-Hƶlder type and applications, Appl. Math. Comput. 219 (2013), 9, 4792ā€“4799. 10.1016/j.amc.2012.10.105Search in Google Scholar

3 G. A. Anastassiou, A new expansion formula, Cubo 5 (2003), 1, 25ā€“31. Search in Google Scholar

4 G. A. Anastassiou, Chebyshevā€“GrĆ¼ss type and comparison of integral means inequalities for the Stieltjes integral, Panamer. Math. J. 17 (2007), 3, 91ā€“109. Search in Google Scholar

5 G. A. Anastassiou, GrĆ¼ss type inequalities for the Stieltjes integral, Nonlinear Funct. Anal. Appl. 12 (2007), 4, 583ā€“593. 10.1142/9789814280792_0025Search in Google Scholar

6 N. S. Barnett, W.-S. Cheung, S. S. Dragomir and A. Sofo, Ostrowski and trapezoid type inequalities for the Stieltjes integral with Lipschitzian integrands or integrators, Comput. Math. Appl. 57 (2009), 2, 195ā€“201. 10.1016/j.camwa.2007.07.021Search in Google Scholar

7 P. Cerone, W. S. Cheung and S. S. Dragomir, On Ostrowski type inequalities for Stieltjes integrals with absolutely continuous integrands and integrators of bounded variation, Comput. Math. Appl. 54 (2007), 2, 183ā€“191. 10.1016/j.camwa.2006.12.023Search in Google Scholar

8 P. Cerone and S. S. Dragomir, New bounds for the three-point rule involving the Riemannā€“Stieltjes integral, Advances in Statistics, Combinatorics and Related Areas, World Science Publishing, River Edge (2002), 53ā€“62. 10.1142/9789812776372_0006Search in Google Scholar

9 P. Cerone and S. S. Dragomir, Approximation of the Stieltjes integral and applications in numerical integration, Appl. Math. 51 (2006), 1, 37ā€“47. 10.1007/s10492-006-0003-0Search in Google Scholar

10 W.-S. Cheung and S. S. Dragomir, Two Ostrowski type inequalities for the Stieltjes integral of monotonic functions, Bull. Aust. Math. Soc. 75 (2007), 2, 299ā€“311. 10.1017/S0004972700039228Search in Google Scholar

11 S. S. Dragomir, Some Inequalities for Riemannā€“Stieltjes integral and applications, Optimization and Related Topics (Ballarat/Melbourne 1999), Appl. Optim. 47, Kluwer, Dordrecht (2001), 197ā€“235. 10.1007/978-1-4757-6099-6_13Search in Google Scholar

12 S. S. Dragomir, Inequalities of GrĆ¼ss type for the Stieltjes integral and applications, Kragujevac J. Math. 26 (2004), 89ā€“122. Search in Google Scholar

13 S. S. Dragomir, Inequalities for Stieltjes integrals with convex integrators and applications, Appl. Math. Lett. 20 (2007), 2, 123ā€“130. 10.1016/j.aml.2006.02.027Search in Google Scholar

14 S. S. Dragomir, Approximating the Riemannā€“Stieltjes integral by a trapezoidal quadrature rule with applications, Math. Comput. Modelling 54 (2011), 1ā€“2, 243ā€“260. 10.1016/j.mcm.2011.02.006Search in Google Scholar

15 S. S. Dragomir, Ostrowski's type inequalities for complex functions defined on unit circle with applications for unitary operators in Hilbert spaces, RGMIA Res. Rep. Coll. 16 (2013), Article ID 6. 10.5817/AM2015-4-233Search in Google Scholar

16 S. S. Dragomir, C. Buşe, M. V. Boldea and L. Braescu, A generalization of the trapezoidal rule for the Riemannā€“Stieltjes integral and applications, Nonlinear Anal. Forum 6 (2001), 2, 337ā€“351. Search in Google Scholar

17 S. S. Dragomir and I. A. Fedotov, An inequality of GrĆ¼ss' type for Riemannā€“Stieltjes integral and applications for special means, Tamkang J. Math. 29 (1998), 4, 287ā€“292. 10.5556/j.tkjm.29.1998.4257Search in Google Scholar

18 S. S. Dragomir and I. A. Fedotov, A GrĆ¼ss type inequality for mappings of bounded variation and applications to numerical analysis, Nonlinear Funct. Anal. Appl. 6 (2001), 3, 425ā€“438. Search in Google Scholar

19 G. Helmberg, Introduction to Spectral Theory in Hilbert Space, North-Holland Ser. Appl. Math. Mech. 6, North-Holland, Amsterdam, 1969. Search in Google Scholar

20 Z. Liu, Refinement of an inequality of GrĆ¼ss type for Riemannā€“Stieltjes integral, Soochow J. Math. 30 (2004), 4, 483ā€“489. Search in Google Scholar

Received: 2014-6-19
Accepted: 2014-9-1
Published Online: 2016-3-24
Published in Print: 2016-6-1

Ā© 2016 by De Gruyter

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