Abstract
Here we present the right and left Riemann–Liouville fractional fundamental theorems of fractional calculus without any initial conditions for the first time. Then we establish a Riemann–Liouville fractional Polya type integral inequality with the help of generalised right and left Riemann–Liouville fractional derivatives. The amazing fact here is that we do not need any boundary conditions as the classical Polya integral inequality requires. We extend our Polya inequality to Choquet integral setting. See also [2].
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
G.A. Anastassiou, Intelligent Mathematics: Computational Analysis (Springer, Heidelberg, 2011)
G.A. Anastassiou, Riemann-Liouville fractional fundamental theorem of Calculus and Riemann-Liouville Fractional Polya type integral inequality and its extension to Choquet integral setting, Bulletin of Korean Mathematical Society (2019)
G. Choquet, Theory of capacities. Ann. Inst. Fourier (Grenoble) 5, 131–295 (1953)
D. Denneberg, Non-additive Measure and Integral (Kluwer Academic Publishers, Boston, 1994)
I. Podlubny, Fractional Differentiation Equations (Academic Press, San Diego, 1999)
G. Polya, Ein Mittelwertsatz für Funktionen mehrerer Veränderlichen. Tohoku Math. J. 19, 1–3 (1921)
G. Polya, G. Szegö, Aufgaben und Lehrsätze aus der Analysis, vol. I (Springer, Berlin, 1925). (German)
G. Polya, G. Szegö, Problems and Theorems in Analysis, vol. I, Classics in Mathematics (Springer, Berlin, 1972)
G. Polya, G. Szegö, Problems and Theorems in Analysis, vol. I, Chinese edn (1984)
F. Qi, Polya type integral inequalities: origin, variants, proofs, refinements, generalizations, equivalences, and applications. RGMIA, Res. Rep. Coll., article no. 20, vol. 16 (2013). http://rgmia.org/v16.php
M. Sugeno, A note on derivatives of functions with respect to fuzzy measures. Fuzzy Sets Syst. 222, 1–17 (2013)
E.T. Whittaker, G.N. Watson, A Course in Modern Analysis (Cambridge University Press, Cambridge, 1927)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Anastassiou, G.A. (2020). Riemann–Liouville Fractional Fundamental Theorem of Calculus and Riemann–Liouville Fractional Polya Integral Inequality and the Generalization to Choquet Integral Case. In: Intelligent Analysis: Fractional Inequalities and Approximations Expanded. Studies in Computational Intelligence, vol 886. Springer, Cham. https://doi.org/10.1007/978-3-030-38636-8_17
Download citation
DOI: https://doi.org/10.1007/978-3-030-38636-8_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-38635-1
Online ISBN: 978-3-030-38636-8
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)