Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

OSTROWSKI TYPE INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS ON SEGMENTS IN LINEAR SPACES

  • Kikianty, Eder (RESEARCH GROUP OF MATHEMATICAL INEQUALITIES AND APPLICATIONS SCHOOL OF ENGINEERING AND SCIENCE VICTORIA UNIVERSITY) ;
  • Dragomir, Sever S. (RESEARCH GROUP OF MATHEMATICAL INEQUALITIES AND APPLICATIONS SCHOOL OF ENGINEERING AND SCIENCE VICTORIA UNIVERSITY) ;
  • Cerone, Pietro (RESEARCH GROUP OF MATHEMATICAL INEQUALITIES AND APPLICATIONS SCHOOL OF ENGINEERING AND SCIENCE VICTORIA UNIVERSITY)
  • Published : 2008.11.30

Abstract

An Ostrowski type inequality is developed for estimating the deviation of the integral mean of an absolutely continuous function, and the linear combination of its values at k + 1 partition points, on a segment of (real) linear spaces. Several particular cases are provided which recapture some earlier results, along with the results for trapezoidal type inequalities and the classical Ostrowski inequality. Some inequalities are obtained by applying these results for semi-inner products; and some of these inequalities are proven to be sharp.

Keywords

File

Download PDF

References

  1. C. D. Aliprantis and O. Burkinshaw, Principles of Real Analysis, North-Holland Publishing Co., New York, 1981
  2. G. A. Anastassiou, Ostrowski type inequalities, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3775-3781
  3. N. S. Barnett, C. Bu¸se, P. Cerone, and S. S. Dragomir, Ostrowski's inequality for vectorvalued functions and applications, Comput. Math. Appl. 44 (2002), no. 5-6, 559-572 https://doi.org/10.1016/S0898-1221(02)00171-2
  4. P. Cerone, Three point rules in numerical integration, Nonlinear Anal. 47 (2001), no. 4, 2341-2352 https://doi.org/10.1016/S0362-546X(01)00358-3
  5. P. Cerone, A new Ostrowski type inequality involving integral means over end intervals, Tamkang J. Math. 33 (2002), no. 2, 109-118
  6. P. Cerone, On relationships between Ostrowski, trapezoidal and Chebychev identities and inequalities, Soochow J. Math. 28 (2002), no. 3, 311-328
  7. P. Cerone and S. S. Dragomir, Three point identities and inequalities for n-time differentiable functions, SUT J. Math. 36 (2000), no. 2, 351-383
  8. P. Cerone and S. S. Dragomir, On some inequalities arising from Montgomery's identity (Montgomery's identity), J. Comput. Anal. Appl. 5 (2003), no. 4, 341-367 https://doi.org/10.1023/A:1024575230516
  9. P. Cerone, S. S. Dragomir, and J. Roumeliotis, Some Ostrowski type inequalities for n-time differentiable mappings and applications, Demonstratio Math. 32 (1999), no. 4, 697-712
  10. S. S. Dragomir, The Ostrowski integral inequality for mappings of bounded variation, Bull. Austral. Math. Soc. 60 (1999), no. 3, 495-508 https://doi.org/10.1017/S0004972700036662
  11. S. S. Dragomir, A generalization of Ostrowski integral inequality for mappings whose derivatives belong to L1[a, b] and applications in numerical integration, J. Comput. Anal. Appl. 3 (2001), no. 4, 343-360 https://doi.org/10.1023/A:1012050307412
  12. S. S. Dragomir, A generalization of the Ostrowski integral inequality for mappings whose derivatives belong to Lp[a, b] and applications in numerical integration, J. Math. Anal. Appl. 255 (2001), no. 2, 605-626 https://doi.org/10.1006/jmaa.2000.7300
  13. S. S. Dragomir, An inequality improving the first Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), no. 2, Article 31, 8 pp
  14. S. S. Dragomir, An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), no. 3, Article 35, 8 pp
  15. S. S. Dragomir, An Ostrowski like inequality for convex functions and applications, Rev. Mat. Complut. 16 (2003), no. 2, 373-382
  16. S. S. Dragomir, Semi-inner Products and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2004
  17. S. S. Dragomir, Ostrowski type inequalities for functions defined on linear spaces and applications for semi-inner products, J. Concr. Appl. Math. 3 (2005), no. 1, 91-103
  18. S. S. Dragomir, An Ostrowski type inequality for convex functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 16 (2005), 12-25
  19. S. S. Dragomir and T. M. Rassias, Generalisations of the Ostrowski Inequality and Applications, Kluwer Acad. Publ., Dordrecht, 2002
  20. C. F. Dunkl and K. S. Williams, Mathematical notes: a simple norm inequality, Amer. Math. Monthly 71 (1964), no. 1, 53-54 https://doi.org/10.2307/2311304
  21. E. Kikianty, S. S. Dragomir, and P. Cerone, Sharp inequalities of Ostrowski type for convex functions defined on linear spaces and applications, Comput. Math. Appl., to appear https://doi.org/10.1016/j.camwa.2008.03.059
  22. W. A. Kirk and M. F. Smiley, Mathematical notes: another characterization of inner product spaces, Amer. Math. Monthly 71 (1964), no. 8, 890-891 https://doi.org/10.2307/2312400
  23. P. R. Mercer, The Dunkl-Williams inequality in an inner product space, Math. Inequal. Appl. 10 (2007), no. 2, 447-450
  24. D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and its Applications (East European Series), vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1991
  25. A. Ostrowski, Uber die Absolutabweichung einer differentiebaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv. 10 (1938), 226-227 https://doi.org/10.1007/BF01214290
  26. T. C. Peachey, A. Mcandrew, and S. S. Dragomir, The best constant in an inequality of Ostrowski type, Tamkang J. Math. 30 (1999), no. 3, 219-222
  27. J. E. Pecaric and S. S. Dragomir, A generalization of Hadamard's inequality for isotonic linear functionals, Rad. Mat. 7 (1991), no. 1, 103-107
  28. H. L. Royden, Real Analysis, second ed., Macmillan Publishing Co. Inc., New York, 1968

Cited by

  1. Asymptotic expressions for error terms of the perturbed mid-point and trapezoid rules vol.15, pp.6, 2012, https://doi.org/10.1080/09720502.2012.10700811
  2. A Unified Generalization of Perturbed Mid-Point and Trapezoid Inequalities and Asymptotic Expressions for Its Error Term vol.0, pp.0, 2014, https://doi.org/10.2478/aicu-2014-0044