INEQUALITIES FOR THE RIEMANN-STIELTJES INTEGRAL OF PRODUCT INTEGRATORS WITH APPLICATIONS

다국어 초록 (Multilingual Abstract)

We show amongst other that if f,g : [a,b] → C are two func- tions of bounded variation and such that the Riemann-Stieltjes integral ∫ba f(t)dg (t) exists, then for any continuous functions h : [a,b] → C, the Riemann-Stieltjes integral ∫ba h(t)d(f(t)g(t)) exists and ∫ba h(t)d(f(t)g(t)) = ∫bah9t)f(t)d(g(t)) + ∫ba h(t)g(t)d(f(t)). Using this identity we then provide sharp upper bounds for the quantity |∫ba h(t)d(f(t)g(t))| and apply them for trapezoid and Ostrowski type inequalities. Some applications for continuous functions of selfadjoint operators on complex Hilbert spaces are given as well.

참고문헌 (Reference)

1 G. A. Anastassiou, "Univariate right fractional Ostrowski inequalities" 14 (14): 1-7, 2012

2 G. A. Anastassiou, "Univariate Ostrowski inequalities" 135 (135): 175-189, 2002

3 P. Cerone, "Trapezoidal-type rules from an inequalities point of view, Handbook of analytic-computational methods in applied mathematics" Chapman & Hall/CRC 65-134, 2000

4 S. S. Dragomir, "The unified treatment of trapezoid, Simpson, and Ostrowski type inequality for monotonic mappings and applications" 31 (31): 61-70, 2000

5 A. M. Acu, "The mean value theorems and inequalities of Ostrowski type" 21 (21): 5-16, 2011

6 S. S. Dragomir, "The Ostrowski integral inequality for mappings of bounded variation" 60 (60): 495-508, 1999

7 W. T. Sulaiman, "Some new Ostrowski type inequalities" 7 (7): 102-107, 2012

8 Z. Liu, "Some inequalities of perturbed trapezoid type" 7 (7): 9-, 2006

9 S. S. Dragomir, "Some inequalities of midpoint and trapezoid type for the Riemann-Stieltjes integral" 47 (47): 2333-2340, 2001

10 S. S. Dragomir, "Some inequalities for continuous functions of selfadjoint operators in Hilbert spaces"

11 Z. Liu, "Some companions of an Ostrowski type inequality and applications" 10 (10): 12-, 2009

12 J. Park, "Some Ostrowskilike type inequalities for differentiable real (α ,m)-convex mappings" 61 (61): 75-91, 2012

13 Z. Liu, "Some Ostrowski type inequalities and applications" 37 (37): 15-22, 2009

14 Y.-X. Xiao, "Remarks on Ostrowskilike inequalities" 219 (219): 1158-1162, 2012

15 S. S. Dragomir, "Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation" 91 (91): 450-460, 2008

16 N. S. Barnett, "Perturbed version of a general trapezoid inequality" Nova Sci. Publ. 1-12, 2003

17 N. Ujevic, "Perturbed trapezoid and mid-point inequalities and applications" 29 (29): 249-257, 2003

18 M. W. Alomari, "Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense" 23 (23): 1071-1076, 2010

19 G. A. Anastassiou, "Ostrowski type inequalities" 123 (123): 3775-3781, 1995

20 G. A. Anastassiou, "Ostrowski inequalities for cosine and sine operator functions" 64 (64): 336-346, 2012

21 N. S. Barnett, "Ostrowski and trapezoid type inequalities for the Stieltjes integral with Lipschitzian integrands or integrators" 57 (57): 195-201, 2009

22 S. S. Dragomir, "On trapezoid inequality via a Gruss type result and applications" 31 (31): 193-201, 2000

23 S. S. Dragomir, "On the trapezoid quadrature formula and applications" 23 : 25-36, 2001

24 S. S. Dragomir, "On the trapezoid inequality for the Riemann-Stieltjes integral with Holder continuous integrands and bounded variation integrators" Nova Sci. Publ. 71-79, 2007

25 Y. Wu, "On the optimal constants of Ostrowskilike inequalities involving n knots" 219 (219): 7789-7794, 2013

26 S. S. Dragomir, "On the midpoint quadrature formula for mappings with bounded variation and applications" 22 : 13-19, 2000

27 J. Park, "On the Ostrowskilike type integral inequalities for mappings whose second derivatives are s∗-convex" 67 (67): 21-35, 2012

28 S. S. Dragomir, "On the Ostrowski's integral inequality for mappings with bounded variation and applications" 4 (4): 59-66, 2001

29 M. Z. Sarikaya, "On the Ostrowski type integral inequality" 79 (79): 129-134, 2010

30 A. Mcd. Mercer, "On perturbed trapezoid inequalities" 7 (7): 7-, 2006

31 A. M. Acu, "On an inequality of Ostrowski type" 2011 (2011): 281-287, 2011

32 Nenad Ujevic, "On Perturbed Mid-point and Trapezoid Inequalities and Applications" 경북대학교 자연과학대학 수학과 43 (43): 327-334, 2003

33 P. Cerone, "On Ostrowski type inequalities for Stieltjes integrals with absolutely continuous integrands and integrators of bounded variation" 54 (54): 183-191, 2007

34 F. Ahmad, "New weighted Ostrowski and Cebysev type inequalities" 71 (71): 1408-1412, 2009

35 Z. Liu, "New sharp bound for a general Ostrowski type inequality" 26 (26): 53-59, 2010

36 B. G. Pachpatte, "New inequalities of Ostrowski and Trapezoid type for n-time differentiable functions" 대한수학회 41 (41): 633-639, 2004

37 W.-J. Liu, "New generalization of perturbed trapezoid, midpoint inequalities and applications" 41 (41): 761-768, 2007

38 P. Cerone, "Midpoint-type rules from an inequalities point of view, Handbook of analytic-computational methods in applied mathematics" Chapman & Hall/CRC 135-200, 2000

39 George A. Anastassiou, "MULTIVARIATE RIGHT FRACTIONAL OSTROWSKI INEQUALITIES" 한국전산응용수학회 30 (30): 445-454, 2012

40 G. Helmberg, "Introduction to Spectral Theory in Hilbert Space" John Wiley & Sons, Inc. 1969

41 K.-L. Tseng, "Improvements of the Ostrowski integral inequality for mappings of bounded variation II" 218 (218): 5841-5847, 2012

42 K.-L. Tseng, "Improvements of the Ostrowski integral inequality for mappings of bounded variation I" 217 (217): 2348-2355, 2010

43 S. S. Dragomir, "Improvements of Ostrowski and generalised trapezoid inequality in terms of the upper and lower bounds of the first derivative" 34 (34): 213-222, 2003

44 P. R. Mercer, "Hadamard's inequality and trapezoid rules for the Riemann-Stieltjes integral" 344 (344): 921-926, 2008

45 A. I. Kechriniotis, "Generalizations of the trapezoid inequalities based on a new mean value theorem for the remainder in Taylor's formula" 7 (7): 13-, 2006

46 N. Ujevic, "Error inequalities for a generalized trapezoid rule" 19 (19): 32-37, 2006

47 Z. Liu, "A sharp general Ostrowski type inequality" 83 (83): 189-209, 2011

48 N. S. Barnett, "A perturbed trapezoid inequality in terms of the third derivative and applications" Nova Sci. Publ. 1-11, 2007

49 N. S. Barnett, "A perturbed trapezoid inequality in terms of the fourth derivative" 9 (9): 45-60, 2002

50 Q. Wu, "A note to Ujevic's generalization of Ostrowski's inequality" 18 (18): 657-665, 2005

51 X.-L. Cheung, "A note on the perturbed trapezoid inequality" 3 (3): 7-, 2002

52 S. W. Vong, "A note on some Ostrowski-like type inequalities" 62 (62): 532-535, 2011

53 B. G. Pachpatte, "A note on a trapezoid type integral inequality" 49 : 85-90, 2004

54 H. Gunawan, "A note on Dragomir-McAndrew's trapezoid inequalities" 33 (33): 241-244, 2002

55 M. Masjed-Jamei, "A new generalization of the Ostrowski inequality and applications" 25 (25): 115-123, 2011

56 P. Cerone, "A generalized trapezoid inequality for functions of bounded variation" 24 (24): 147-163, 2000

57 M. W. Alomari, "A companion of Ostrowski's inequality with applications" 3 (3): 9-14, 2011

58 N. S. Barnett, "A companion for the Ostrowski and the generalised trapezoid inequalities" 50 (50): 179-187, 2009

59 Zheng Liu, "A NOTE ON OSTROWSKI TYPE INEQUALITIES RELATED TO SOME s-CONVEX FUNCTIONS IN THE SECOND SENSE" 대한수학회 49 (49): 775-785, 2012