SOME GRÜSS TYPE INEQUALITIES FOR THE RIEMANN-STIELTJES INTEGRAL WITH LIPSCHITZIAN INTEGRATORS

Year 2014, Volume: 2 Issue: 1, 36 - 44, 01.06.2014

M.w. Alomarı S.S.DRAGOMIR

https://izlik.org/JA69GX78UK

Abstract

In this paper several new inequalities of Gr¨uss’ type for the Riemann–Stieltjes integral with Lipschitzian integrators are proved

Keywords

Gr¨uss inequality , Function of bounded variation , H¨older continuousfunctions

References

• M.W. Alomari and S.S. Dragomir, New Gr¨uss type inequalities for the Riemann–Stieltjes integral with monotonic integrators and applications, submitted. Avalibale at: M.W. Alomari, Some Gr¨uss type inequalities for Riemann-Stieltjes integral and applications, Acta Mathematica Universitatis Comenianae, (N.S.) 81 (2012), no. 2, 211–220.
• G.A. Anastassiou, Gr¨uss type inequalities for the Stieltjes integral. Nonlinear Funct. Anal. Appl. 12 (2007), no. 4, 583–593.
• G.A. Anastassiou, Chebyshev-Gr¨uss type and comparison of integral means inequalities for the Stieltjes integral. Panamer. Math. J. 17 (2007), no. 3, 91–109.
• P. Cerone and S.S. Dragomir, New bounds for the ˇCebyˇsev functional, Appl. Math. Lett. 18 (2005) 603–611.
• P. Cerone and S.S. Dragomir, A refinement of the Gr¨uss inequality and applications, Tamkang J. Math. 38 (1) (2007) 37–49.
• S.S. Dragomir, Inequalities of Gr¨uss type for the Stieltjes integral and applications, Kragu- jevac J. Math. 26 (2004) 89–112.
• S.S. Dragomir, New Gr¨uss’ type inequalities for functions of bounded variation and applica- tions, Appl. Math. Lett. In press, http://dx.doi.org/10.1016/j.aml.2011.12.027, (2012).
• S.S. Dragomir, New estimates of the ˇCebyˇsev functional for Stieltjes integrals and applica- tions, J. Korean Math. Soc., 41(2) (2004), 249–264.
• S.S. Dragomir, Sharp bounds of ˇCebyˇsev functional for Stieltjes integrals and applications, Bull. Austral. Math. Soc., 67(2) (2003), 257–266.
• S.S. Dragomir, Some integral inequality of Gr¨uss type, Indian J. Pure Appl. Math., 31(4) (2000), 397–415.
• S.S. Dragomir, Gr¨uss type integral inequality for mappings of r-H¨older type and applications for trapezoid formula, Tamkang J. Math., 31(1) (2000), 43–47.
• S.S. Dragomir and I. Fedotov, An inequality of Gruss type for Riemann–Stieltjes integral and applications for special means, Tamkang J. Math., 29(4) (1998), 287–292.
• G. Gr¨uss,Uber das maximum des absoluten Betrages von ¨ 1 b−aaf (x) g (x) dx − bf (x) g (x) dx − 1 (b−a)2a ag (x) dx, Math. Z. 39 (1935) 215–226.
• Z. Liu, Refinement of an inequality of Gr¨uss type for Riemann-Stieltjes integral. Soochow J. Math. 30 (2004), no. 4, 483–489. 1Department of Mathematics, Faculty of Science and Technology, Jadara Univer- sity, 21110 Irbid, Jordan.
• E-mail address: [e-mail] 2Mathematics, School of Engineering & Science, Victoria University, PO Box 14428,
• Melbourne City, MC 8001, Australia. 3School of Computational & Applied Mathematics, University of the Witwater- srand, Johannesburg, South Africa.
• E-mail address: [e-mail]