Document Type : Research Paper


1 Department of Mathematics, Samarkand State University, University boulevard 15, Samarkand 140104, Uzbekistan.

2 Department of Mathematics, Dawood University of Engineering and Technology, New M. A. Jinnah Road, Karachi-74800, Pakistan.


We apply the Riemann-Liouville fractional integral to generalize a companion of Ostrowski's type integral inequality. The present article recaptures all the results of M. W. Alomari's article and also for one more article of different authors. Applications are also deduced for numerical integration, probability theory and special means.


Main Subjects

1. M. Alomari and M. Darus, Some Ostrowski type inequalities for quasi-convex functions with applications to special means, RGMIA, Preprint 13 (2) (2010) Art. 3, pp. 1-9.
2. M. Alomar, M. Darus, S.S Dragomir and P. Cerone, Ostrowski type inequalities for functions whose derivatives are s–convex in the second sense, Appl. Math. Lett., 23 (2010), pp. 1071-1076.
3. M.W. Alomari, A companion of Ostrowski’s inequality with applications, Trans. J. Math. Mech., 3 (1) (2011), pp. 9-14.
4. G.A. Anastassiou,Multivariate Ostrowski type inequalities, Acta. Math. Hungar., 76 (1997), pp. 267-278.
5. G. Anastassiou, M.R. Hooshmandasl, A. Ghasemi and F. Muftakharzadeh, Montgomery Identities for Fractional Integrals and Related Fractional Inequalities, J. Inequal. Pure Appl. Math., 10 (4) (2009), Art. 97, pp. 6.
6. L. Avazpour, Fractional Ostrowski Type Inequalities for Functions whose Derivatives are Prequasinvex, J. Inequal. Spec. Funct., 9 (2) (2018), pp. 15-29.
7. N.S. Barnett, S.S. Dragomir and I. Gomma, A companion for the Ostrowski and the generalised trapezoid inequalities, J. Mathematical and Computer Modelling, 50 (2009), pp. 179–187.
8. X.L. Cheng, Improvement of some Ostrowski-Grüss type inequalities, Comput. Math. Appl., 42 (1/2) (2001), pp. 109-114.
9. S.S. Dragomir and S. Wang, An inequality of Ostrowski-Grüss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Comput. Math. Appl., 33 (11) (1997), pp. 16-20.
10. S.S. Dragomir, P. Cerone and J. Roumeliotis, A new Generalization of Ostrowski Integral Inequality for Mappings whose Derivatives are Bounded and Applications in Numerical Integration and for Special Means, Appl. Math. Lett., 13 (2000), pp. 19-25.
11. S.S. Dragomir and T.M. Rassias, Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academics Publishers, Dordrecht, 2002.
12. S.S. Dragomir, A companion of Ostrowski’s inequality for functions of bounded variation and applications, RGMIA Preprint, 5 (2002) Art. 28.
13. S.S. Dragomir, Some companions of Ostrowski’s inequality for absolutely continuous functions and applications, Bull. Korean Math. Soc., 42 (2) (2005), pp. 213-230.
14. S.S. Dragomir, A.R. Khan, M. Khan, F. Mehmood and M.A. Shaikh, A new integral version of generalized Ostrowski-Grüss type inequality with applications, Journal of King Saud University – Science, 34 (2022), pp. 1-6.
15. R. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, Springer Verlag, Wien and New York, 1997.
16. G. Grüss, Über das Maximum des absoluten Betrages von, Math. Z., 39 (1) (1935), pp. 215-226.
17. A. Guessab and G. Schmeisser, Sharp integral inequalities of the Hermite-Hadamard type, J. Approx. Th., 115 (2002), pp. 260-288.
18. A. Hassan, A.R. Khan, F. Mehmood and M. Khan, Fuzzy Ostrowski Type Inequalities Via $\phi-\lambda-$−Convex Functions, J. Math. Computer Sci., 28 (2023), pp. 224-235.
19. A. Hassan, A.R. Khan, F. Mehmood and M. Khan, Fuzzy Ostrowski Type Inequalities Via h-Convex, J. Math. Comput. Sci., 12 (2022), pp. 1-15.
20. A. Hassan, A.R. Khan, F. Mehmood and M. Khan, BF-Ostrowski Type Inequalities Via $\phi-\lambda-$−Convex Functions, International Journal of Computer Science and Network Security, 21 (10) (2021), pp. 177-183.
21. Z. Liu, Some companions of an Ostrowski type inequality and applications, J. Ineq. Pure & Appl. Math., 10 (2) (2009), Art. 52, pp. 12.
22. M. Matic, J.E. Pečarić and N. Ujevic, Improvement and further generalization of inequalities of Ostrowski-Grüss type, Comput. Math. Appl., 39 (3/4) (2000), pp. 161-175.
23. F. Mehmood, K. Saleem, Z.A. Naveed, G.M. Khan and A. Rahman, A Companion of Weighted Ostrowski’s Type Inequality and Applications to Numerical Integration, Global Journal of Pure and Applied Mathematics, 16 (4) (2020), pp. 577-586.
24. F. Mehmood, A.R. Khan, M.A. Shaikh and M.W. Alomari, Generalisation of Companion of Ostrowski’s Type Inequality Via Riemann-Liouville Fractional Integral for Mappings whose 1st Derivatives are Bounded with Applications, Kragujevac Journal of Mathematics, to appear.
25. F. Mehmood and A. Soleev, Generalization of Ostrowski’s Type Inequality Via Riemann-Liouville Fractional Integral and Applications in Numerical Integration, Probability Theory and Special Means, Jordan Journal of Mathematics and Statistics, to appear.
26. G.V. Milovanovic and J.E. Pečarić, On generalizations of the inequality of A. Ostrowski and some related applications, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., (1976), pp. 544- 576, 155-158.
27. D.S. Mitrinović , J.E. Pečarić and A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
28. F. Nawaz, Z.A. Naveed, F. Mehmood, G.M. Khan and K. Saleem, A Companion of Weighted Ostrowski’s type Inequality for Functions whose 1st Derivatives are Bounded with Applications, Global Journal of Pure and Applied Mathematics, 16 (4) (2020), pp. 515-522.
29. A.M. Ostrowski, ¨U ber die Absolutabweichung einer Differentiebaren Funktion von Ihren Integralmittelwert, Comment. Math. Helv., 10 (1938), pp. 226-227.
30. M.Z. Sarikaya, H. Filiz and M.E. Kiris, On Some Generalized Integral Inequalities for Riemann-Liouville Fractional Integrals, Filomat, 29 (6) (2015), pp. 1307-1314.
31. M.Z. Sarikaya, H. Yaldiz and N. Basak, New Fractional Inequalities of Ostrowski Grüss Type, Le Matematiche, 69 (1) (2014), pp. 227-235.
32. M.Z. Sarikaya and H. Ogunmez, On New Inequalities Via Riemann Liouville Fractional Integration, Abst. Appl. Anal., 2012 (2012), Art. 428983, pp. 10.
33. M.A. Shaikh, A.R. Khan and F. Mehmood, Estimates for weighted Ostrowski-Grüss type inequalities with applications, Analysis – De Gruyter, 2022, pp. 1-11.
34. M.A. Shaikh, F. Mehmood and A.R. Khan, Two–Point Ostrowski Type Inequality with Parameter, Submitted.
35. F. Zafar, Some Generalizations of Ostrowski Inequalities and Their Applications to Numerical Integration and Special means, Unpublished doctoral thesis, Bahauddin Zakariya University, Multan, Pakistan, 2010.