Abstract
Here we present the most general fractional representation formulae for a function in terms of the most general fractional integral operators due to Kalla [4–6]. The last include most of the well-known fractional integrals such as of Riemann-Liouville, Erdé lyi-Kober and Saigo, etc. Based on these we derive very general fractional Ostrowski type inequalities. It follows [2].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
G. Anastassiou, M. Hooshmandasl, A. Ghasemi, F. Moftakharzadeh, Montgomery identities for fractional integrals and related fractional inequalities. J. Inequal. Pure Appl. Math. 10(4), article no 97, 6 (2009)
G.A. Anastassiou, Most general fractional representation formula for functions and implications. Serelica Math. 40, 89–98 (2014)
E.A. Chistova (originator), Hypergeometric function, Encyclopedia of Mathematics, http://www.encyclopediaofmath.org/index.php?title=Hypergeometric_function&oldid=12873, ISBN 1402006098
S.L. Kalla, On operators of fractional integration. I. Mat. Notae 22, 89–93 (1970)
S.L. Kalla, On operators of fractional integration. II Mat. Notae 25, 29–35 (1976)
S.L. Kalla, Operators of fractional integration. In: Proceedings of Conference on Analytic Functions, Kozubnik 1979, Lecture Notes of Mathematic, vol. 798, pp. 258–280 (Springer, Berlin, 1980)
V.S. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics Series, vol. 301 (Longman Scientific & Technical, Harlow, 1994)
V.S. Kiryakova, All the special functions are fractional differintegrals of elementary functions. J. Phys. A: Math. Gen. 30, 5085–5103 (1997)
V.S. Kiryakova, On two Saigo’s fractional integral operators in the class of univalent functions. Fractional Calculus & Applied Analysis 9(2), 159–176 (2006)
S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, USA, 1993)
D.S. Mitrinovic, J.E. Pecaric, A.M. Fink, Inequalities for functions and their integrals and derivatives (Kluwer Academic Publishers, Dordrecht, 1994)
R.K. Raina, Solution of Abel-type integral equation involving the Appell hypergeometric function. Integr. Transform. Spec. Funct. 21(7), 515–522 (2010)
M. Saigo, A remark on integral operators involving the Gauss hypergeometric functions. Math. Rep. Kyushu Univ. 11, 135–143 (1978)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Anastassiou, G.A. (2016). The Most General Fractional Representation Formula for Functions and Consequences. In: Intelligent Comparisons: Analytic Inequalities. Studies in Computational Intelligence, vol 609. Springer, Cham. https://doi.org/10.1007/978-3-319-21121-3_20
Download citation
DOI: https://doi.org/10.1007/978-3-319-21121-3_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-21120-6
Online ISBN: 978-3-319-21121-3
eBook Packages: EngineeringEngineering (R0)