Ostrowski type inequalities via the Katugampola fractional integrals

Mustafa Gürbüz; Yakup Taşdan; Erhan Set; Mustafa Gürbüz; Yakup Taşdan; Erhan Set

doi:10.3934/math.2020004

AIMS Mathematics

2020, Volume 5, Issue 1: 42-53. doi: 10.3934/math.2020004

Previous Article Next Article

Research article

Ostrowski type inequalities via the Katugampola fractional integrals

1.
Ağrı İbrahim Çeçen University, Department of Mathematics, Faculty of Education, Ağrı, Turkey
2.
Ağrı İbrahim Çeçen University, Institute of Science, Ağrı, Turkey
3.
Ordu University, Faculty of Science and Letters, Department of Mathematics, Ordu, Turkey

Received: 15 July 2019 Accepted: 26 September 2019 Published: 17 October 2019
MSC : 26A33, 26D10, 26D15

The main aim of this study is to reveal new generalized-Ostrowski-type inequalities using Katugampola fractional integral operator which generalizes Riemann-Liouville and Hadamard fractional integral operators into a single form. For this purpose, at first, a new fractional integral identity is generated by the researchers. Then, by using this identity, some inequalities for the class of functions whose certain powers of absolute values of derivatives are $p-$convex are derived. Some applications to special means for positive real numbers are also given. It is observed that the obtained inequalities are generalizations of some well known results.
- Katugampola fractional integral,
- Ostrowski type inequalities,
- p-convex functions
Citation: Mustafa Gürbüz, Yakup Taşdan, Erhan Set. Ostrowski type inequalities via the Katugampola fractional integrals[J]. AIMS Mathematics, 2020, 5(1): 42-53. doi: 10.3934/math.2020004

Related Papers:

Abstract

The main aim of this study is to reveal new generalized-Ostrowski-type inequalities using Katugampola fractional integral operator which generalizes Riemann-Liouville and Hadamard fractional integral operators into a single form. For this purpose, at first, a new fractional integral identity is generated by the researchers. Then, by using this identity, some inequalities for the class of functions whose certain powers of absolute values of derivatives are $p-$convex are derived. Some applications to special means for positive real numbers are also given. It is observed that the obtained inequalities are generalizations of some well known results.

References

[1]	R. P. Agarwal, M. J. Luo, R. K. Raina, On Ostrowski type inequalities, Fasciculi Mathematici, 56 (2016), 5-27. doi: 10.1515/fascmath-2016-0001
[2]	M. Alomari, M. Darus, S. S. Dragomir, et al. Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense, Appl. Math. Lett., 23 (2010), 1071-1076. doi: 10.1016/j.aml.2010.04.038
[3]	H. Budak, M. Z. Sarikaya, E. Set, Generalized Ostrowski type inequalities for functions whose local fractional derivatives are generalized s-convex in the second sense, Journal of Applied Mathematics and Computational Mechanics, 15 (2016), 11-21.
[4]	A. Erdélyi, W. Magnus, F. Oberhettinger, et al. Higher transcendental functions, Vol. I-Ⅲ, Krieger Pub., Melbourne, Florida, 1981.
[5]	İ. İşcan, Ostrowski type inequalities for p-convex functions, New Trends in Mathematical Sciences, 3 (2016), 140-150.
[6]	U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865.
[7]	U. N. Katugampola, New approach to generalized fractional derivatives, Bulletin of Mathematical Analysis and Applications, 6 (2014), 1-15.
[8]	A. A. Kilbas, Hadamard-type fractional calculus, Journal of Korean Mathematical Society, 38 (2001), 1191-1204.
[9]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 2006.
[10]	G. W. Leibniz, "Letter from Hanover, Germany to G.F.A. L'Hospital, September 30, 1695", Leibniz Mathematische Schriften. Olms-Verlag, Hildesheim, Germany, 1962. p.301-302, First published in 1849.
[11]	A. Ostrowski, Uber die absolutabweichung einer differentienbaren funktionen von ihren Integralmittelwert, Comment. Math. Hel., 10 (1938), 226-227.
[12]	M. Z. Sarikaya, H. Budak, Generalized Ostrowski type inequalities for local fractional integrals, P. Am. Math. Soc., 145 (2017), 1527-1538.
[13]	E. Set, New inequalities of Ostrowski type for mappings whose derivatives are s-Convex in the second sense via fractional integrals, Comput. Math. Appl., 63 (2012), 1147-1154. doi: 10.1016/j.camwa.2011.12.023
[14]	K. S. Zhang, J. P. Wan, p-Convex functions and their properties, Pure Appl. Math., 23 (2007), 130-133.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)